UC-NRLF 


SB    27fl 


S 


MKNT A!,    v  Nl»   "  -MTTJ 


jjrtiljmctxul  <]••• 


PERTAINING  TO   THE  WORK  OF  HANDLING  WHOLE 
AND  FRACTIONAL   HUM^fiS  AND  A  GREAT 
VARIETY  OF  IMPORTANT  PRACTICAL 
'OBLEMS 


BEING  A  COMPLETE  IHTEOE7CTOHY  vVOEZ 


S  (  .'  !  K 


i  377. 


IN  MEMORIAM 
FLOR1AN  CAJORI 


SOULE'S 


EMBRACING   MENTAL  AND  WRITTEN 

jirttijmettcal  (J.xcrdses  *ni  Camples* 

I'KKTAININ<;  TO    Til  K  WORK    OF    HANDLING     NUM  HERS   AND  THK 

APPLICATION  TUKREOF   TO  SIMPLE   PRACTICAL  QUESTIONS 

INVOLVING  THE   PRINCIPLES  OF 

Addition,  Subtraction,  Multiplication  and  Division 

OF  WHOLE  AND  FRACTIONAL  NUMBERS. 

Also  a  great  varii'ty  ami  ;i  large  number  of  important  practical  problems,  such  an 

occur  in  the  ('oiiiitiiigio.mi,  Factory,  Workshop,  on  the  Plantation, 

and  throughout  the  various  departments  of  business  life. 

/)rsi</nctf   ns   a    Siiftjflctticnt   to 

Soule's  Philosophic,  Commercial  and  Exchange  Calculator 

AND,   AS    AN 


S  C  I  E  N  C  E    OF     N  U  M  B  K  R  8  . 

By    G^EO.    SOTJLE, 

•  ! 

Practical  and   Consulting  Accountant,    Commercial  Lawyer,  President   of 

Sony's  Commercial  College  arnf  Literary  Institute,  author  of  "Con- 

tractions in  Numbers"  and  the  "  Analytic  ai;d  Philosophic, 

(  'cwnicrcial  and  Exchange  Calculator." 


NEW  OKLK.L\S  .- 


Autlior. 


1877. 


ENTERED  ACCORDING  TO  ACT  OF  CONGRESS,  IN  THE  YEAR  1877, 
BY     O  E  O  .    S  O  TJ  L  sT, 

IN   THE  OFFICE  OF  THE   LIBRARIAN   OF   CONGRESS,   AT   WASHINGTON. 


•FELINE,  PRINTERS,  ~| 
avier  St.,'  N.  0.          ) 


CLARK  &  HOFELINE,  PRINTERS, 
112  Grav 


PREF  AC  E. 


|HE  design  of  this  work  is  twofold  :  1.  As  a  supplement 
to  the  author's  large  and  advanced  treatise,  which  does 
not  contain  sufficient  primary  work  to  meet  the  wants  of 
young  pupils.  2.  As  an  introductory  treatise  to  the  science  of 
numbers.  It  is  especially  designed  to  supply  the  requirements 
of  primary  and  intermediate  classes,  and,  at  the  same  time,  it 
presents  much  practical  work  of  rare  import  to  the  advanced 
student.  Bills  and  invoices  of  various  forms  for  many  depart- 
ments of  business  constitute  a  special  feature  of  the  work.  It  is 
believed  to  possess  superior  merit  on  the  following  points  :  1 . 
In  the  arrangement  and  character  of  the  mental  exercises  and 
the  logical  methods  of  mental  training.  2.  In  the  extent,  variety* 
practical  and  scientific  character  oflhe  problems.  3.  In  the 
elucidation  of  subjects.  4.  In  the  philosophic  solution  of  prob- 
lems, by  which  system  all  the,  reasoning  organs  of  the  mind 
are  expanded  and  the  learner  capacitated,  not  only  to  produce 
the  results  of  problems,  but  to  observe  fine  distinct!  ms,  reason 
logically,  and  deduce  correctly  ;  thus  qualifying  for  a  high  plane 
of  usefulness  in  various  vocations  of  life. 

The  philosophic  system  is  believed  to  be  the  most  valuable 
improvement  yet  made  to  impart  a  thorough  knowledge  of  the 
principles  of  numbers  and  capacitate  the  learner  to  utilize  the 
same  in  the  practical  affair*'  of  business  life,  and  a*  it  is  the  only 
natural  system,  it  is  destined  at  no  distant  day,  to  be  adopted  by 
all  reasoning  minds  and  efficient  instructors.  But,  notwithstand- 
ing its  superiority  and  the  fact  that  its  advocates  include  manv 


IV  PREFACE. 

of  the  most  profound  mathematical  minds,  yet,  like  every  other 
improvement  or  discovery  in  education,  commerce,  art,  or  science^ 
it  has  opponents  and  is  regarded  with  indifference  by  those  who 
are  satisfied  with  the  non-progressive  and  non-reasoning  methods 
of  past  ages. 

With  an  experience  extending  through  a  period  of  twenty 
years  with  nearly  four  thousand  students,  the  author  has  tested 
the  advantages  of  the  philosophic  system,  and  from  a  full  knowl- 
edge of  its  superior  merits,  he  conscientiously  assures  his  co-labor- 
ers in  the  mathematical  field  of  education  that  a  more  thorough 
knowledge  of  the  science  of  numbers  can  be  imparted,  and  in 
far  less  time,  by  this  system  than  by  the  usual  methods  and  sys- 
tems of  work. 

The  science  of  numbers  is  the  front  door  to  the  grand  Temple 
of  mathematics,  in  which  are  displayed  some  of  the  most  beauti- 
ful principles  of  logic  and  profound  syllogistic,  analogical  and 
axiomatical  truths  to  l>e  found  in  the  vast  fields  of  thought. 
And  all  who  aspire  to  pre-eminence  in  brain  power — all  who 
hope  to  ascend  to  the  highest  planes  of  mathematical  knowledge, 
must  devote  themselves  earnestly  to  this  subject. 

In  the  selection  of  the  material  and  the  elements  for  it  prob- 
lems, this  work  does  not  present  the  toys  and  play  things  of  the 
nursery,  nor  does  it  confine  itself  to  the  articles  bought  and  sold 
on  'change.  Instead  of  gyrating  in  the  non-practical  and  non- 
progressive  paths  described  by  its  hundreds  of  predecessors,  it 
has  diverged  into  new  channels  and  derived  the  facts  and  ele- 
ments of  many  of  its  problems  from  geography,  history  and 
chronology ;  from  educational  and  commercial  statistics ;  from 
Natural  Philosophy,  Astronomy,  Geology  and  Chemistry  ;  from 
Anatomy,  Physiology  and  Hygiene;  and  from  many  other 


PREFACE.  V 

departments  of  scientific  knowledge.  Through  this  means,  the 
work  is  rendered  far  rno^e  interesting,  and  as  it  brings  into  use 
different  organs  of  the  mind  from  those  which  consider  the  com- 
putation of  numbers  only,  it  thereby  imtaes  the  mind  of  the 
learner  with  much  valuable  information  without  the  cost  of 
additional  study  or  the  expenditure  of  additional  time. 

The  work  has  been  prepared  during  such  intervals  of  time 
as  the  author  could  command  from  his  professional  duties  as 
teacher  of  Business  Sciences,  and  Consulting  and  Practical 
Accountant ;  and,  nowithstanding  great  care  has  been  bestowed 
upon  it,  it  is  not  improbable  that  some  typographical  or  other 
errors  may  have  escaped  notice.  Should  any  such  be  found,  the 
author  will  esteem  it  a  favor  to  be  informed  of  them,  in  order 
that  they  may  be  expunged  in  future  editions. 

The  author  avails  himself  of  this  occasion  to  extend  his 
thanks  to  his  associate  instructors  and  advanced  students  for 
their  kindly  aid  in  proof  reading,  and  especially  does  he  express 
his  gratitude  to  his  assistant  instructor,  Mr  B.  I).  Rowlee,  for 
services  cheerfully  rendered  in  proof  reading  and  the  re-working 
of  problems. 

Soliciting  for  the  work  a  thorough  examination  and  a  just 
measure  of  its  merits,  with  the  earnest  hope  that  it  may  prove 
acceptable,  and  be  of  service  in  unfolding  the  principles  of  the 
beautiful  science  of  numbers,  and  aid  in  advancing  the  interests 
of  the  rising  generation,  it  is  now  submitted  to  the  public. 

THE  AUTHOR. 
NEW  ORLEANS,  Jan.  4,  1877. 


TOPICAL    INDEX. 


Pago. 

DEFINITIONS 1  to       5 

FRENCH  SYSTEM  OF  NUMERATION 6 

ENGLISH  SYSTEM  OF  NUMERATION 7 

THE  ROMAN  SYSTEM  OF  NOTATION 8 

TABLE  OF  ROMAN  CHARACTERS 8 

EXERCISES  IN  NOTATION  AND  NUMERATION  9 

ADDITION— (Definitions  and  Introduction.) 10 

Addition  Tables 11  to     12 

Addition  and  Subtraction  Tables 13  to     14 

Signs  and  Abbreviations 15 

Mental  Exercises  in  Addition 10  to     17 

Examples  in  Addition IT  to     23 

Proof  of  Addition 18 

Dollar  and  Cent  Signs 23 

Addition  of  Dollars  and  Cents 23 

Miscellaneous  Examples  in  Addition 24  to     31 

STBTRACTION—  Definitions  and  Introduction 31    to      32 

Oral  Exercises  in  Subtraction 32 

Examples  in  Subtraction 32   to     39 

To  subtract  Dollars  and  Cents 35 

MULTIPLICATION— Definitions  and  Introduction...  40 

Multiplication  Table 41 

Oral  Exercises  in  Multiplication 42   to     43 

To  Multiply  when  the  Multiplier  consists  of 

only  one  figure 43 

Examples  in  Multiplication 43  to     53 

To  Multiply  when  the  Multiplier  consists  of 

more  than  one  figure 45 

To  Multiply  when  either  the  Multiplicand  or 
Multiplier  or  both    have   naughts  on  the 

right 47 

To  Multiply    by   the   Factors   of  a    number  48 
To  Multiply  when  the  Multiplicand  or  Mul- 
tiplier contains  dollars  and  cents 

Miscellaneous  Problems  in  Multiplication...  49  to     53 

DIVISION— Introduction  and    Definition 53 

Principles  of  Division 53   to     54 

Proof  of  Division    »..  54 


TOPICAL    INDEX.  Vll 

DIVISION— Oral  Exercises  in 54  to     55 

Fractional  Numbers  and  Examples 55  to     57 

Written  Exercises 57 

To  Divde  when  the  Divisor  does  not  exceed 

12  57 

Short  Division , 57  to     60 

To  Divide  when  the  Divisor  exceeds  12 59 

Examples  in  Short  Division 58  to     59 

Long  Division 60 

To  divide  when  there  are  naughts  on  the 

right  of  the  Divisor  61 

Examples  in  Long  Division 60  to     70 

To  Divide  by  the  Factors  of  a  number 63 

Problems  involving   the   English    Money    of 

Account 69  to     70 

Miscellaneous  Problems  involving  the  Prin- 
ciples of  Addition,  Subtraction,  Multipli- 
cation and  Division 70  to  72 

CANCELLATION— Fully    explained 72  to     75 

PROPERTIES— and  Definitions  of  Numbers 75  to     77 

Divisibility  of  Numbers 77  to     78 

FRACTIONS— and  Definitions  of  Fractions 78 

Classification  of  Fractions 79  to     80 

General  Principles  of  Fractions 80 

REDUCTION  OF  FRACTIONS 80 

Oral  Exercises  in  Fractions 80  to     86 

Greatest  Common  Divisor — and  Examples...  86  to     87 

Least  Common  Multiple — and  Examples 88 

To  Reduce  Fractions  to  their  lowest  terms..  89 
To    Reduce    Whole   or   Mixed    numbers    to 

Improper  Fractions  90 

To  Reduce  Improper  Fractions  to  Whole  or 

Mixed  numbers 90 

To  Reduce  Compound  Fractions  to  Simple 

Fractions 91 

To  reduce  fractions  of  different  denomina- 
tors to  equivalent  fractions  of  a  common 
denominator  or  of  the  least  common  de- 
nominator   92  to  94 

DENOMINATE  FRACTIONS 94 

To   reduce   a    denominate    fraction   from   a 

greater  unit  to  a  less 94 

To  reduce  a  denominate  fraction  from  a  less 

unit  to  a  greater 94 

ADDITION  OF  FRACTIONS 94  to     98 

SUBTRACTION  OF  FRACTIONS...  99  to   102 


Vlll  TOPICAL    INDEX. 

MULTIPLICATION  OF  FRACTIONS 103  to  111 

To  multiply  Abstract  Fractional  numbers...  108 
Miscellaneous  examples  in  Multiplication  of 

Fractions 108  to   111 

DIVISION  OF  FRACTIONS 112  to   121 

Division  of  Abstract  numbers 116 

Miscellaneous  examples  in  Division  of  Frac- 
tions   117  to  121 

Miscellaneous  examples  involving  the  prin- 
ciples of  Addition,  Subtraction,  Multipli- 
cation and  Division  of  Fractions 121  to  127 

DECIMAL     FRACTIONS—  Definition*     and     Intro- 
duction   127  to   144 

Exercises 131 

Principles 133 

Reduction  of  Decimals 133 

To  reduce  Decimal  Fractions  to  a  common 

denominator 133 

To  reduce  a  decimal  to  a  common  fraction..  133 
To  reduce  common  fractions  to  equivalent 

decimals 134 

ADDITION  OF  DECIMALS l.:<; 

SUBTRACTION  OF  DECIMALS 137 

MULTIPLICATION  OF  DECIMALS 138 

To  multiply  a  decimal  or  mixed  number  by 

10,  100,  1000,  etc 140 

DIVISION  OF  DECIMALS 140 

To   divide   Decimal   Fractions   by    10,    100, 

1000,  etc 143 

MISCELLANEOUS    PRACTICAL    PROBLEMS— 

^uch  as  occur  in  the  counting-room,  factory, 

workshop,  on  the  plantation,  and  in  the 

various  departments  of  business  life 144  to   149 

BILLS  AND  INVOICES 149  to   165 

WEIGHTS  AND  MEASURES 165 

Definitions  and  comparison  of  different  units.  165  to    167 

Tables  of 167  to   173 


DEFINITIONS. 


1.     Definition   is   the   meaning   or  import  of  a  word 
ex  pressed  by  other  words. 

is  classified  knowledge. 


•>.  Quantity  is  nny  thin^  that  can  he  increased  or 
diminished. 

4.  A  I  llit  is  ti  single  tiling  of  whatsoever  denomination 
or  nature. 

5.  A  Number  is  a  unit  or  a  collection  of  units. 

<>.  All  Abstract  Xmilber  is  one  in  which  the  kind  of 
unit  or  (juantity  is  not  designated,  thus:  three,  four,  five, 

etc. 

7.    A  Denominate  or  Concrete  Number  is  one  in 

which  the  kind  of  unit  is  designated.  Thus:  two  pounds, 
five  vards.  nine  dollars,  etc. 

K.  A  Compound  Number  is  a  denominate  numher 
expressed  in  two  or  more  denominations,  thus:  ,">  years.  4 
months  and  S  days:  '1  milrs,  .">  furlongs  and  10  rods:  2 
yards,  '2  feet  and  5  inches. 

!>.    An  Arithmetical  Complement  of  a  Number 

i.sthe  dillerenc<-  hetwern  the  numher  and  a  unit  of  the  next 
higher  nrder,  thus:  .'I  is  the  arithmetical  Complement  of  7  ; 
-(>  is  the  arithmetical  complement  of  71  :  l!>  is  the  arith- 
metical complement  of  J)Sl. 

1(>.  A  Problem  is  a  <|uestion  ju-oposed  or  given  for 
solution. 

11.  Philosophy  —  the  knowledge  of  phenomena  as 
explained  hy  and  resolved  into  causes  and  reasons,  powers 

aud  hi\vs. 


4  Arithmetical  Exercises  and  Examples. 

12.  ^  Arithmetic    is   tin-   Science   of  Numbers:   or  to 
define  it  more  extemledly,  it  is  that  branch  of  .Mathematics 
which   treats  of  the   properties   and    relations  of  numbers 
when  expressed  by  the  aid  of  figures,  either  singly  or  com- 
bined.     These  principles  and  relations  of  numbers  combined 
with  the  facts  relating  to  problems,  aie  applied,  by  the  rea- 
soning powers  of  man  to  the  solution  of  all  numerical  prob- 
lems of  business  affairs  and  practical  life. 

13.  Figures;  in  Arithmetic  _ ///////>  x  are  characters  used 
to  represent  numbers.      The  ten    Arable   figures  which  we 
use,  are 

Nan-lit  or  Cipher    One    T\v.>     Tim-..     F"iir     Fi\i-     Six     Seren     Ki^lit     Nino 

0  1       2       3        4       5      (I       7        8        !) 

By  properly  combining  these  ten  figures  all  possible  num- 
bers may  be  represented. 

The  1,  2,  3,  4,  5,  6,  7,  8,  and  H  are  sometimes  called  dibits. 
They  are  also  called  the  significant  figures  because  each  sig- 
nifies a  number  when  alone. 

The  0  is  so  called  because  by  itself  it  docs  not  signify  or 
represent  any  number.  It  expresses  number  only  when 
used  in  connection  with  other  figures. 

14.  Yalue  of  Figures — Figures  have  two  values,  a 
afin}rfc  and  a  local  value:  thus  when  we  write  2,  independ- 
ent of  other  figures,  it  has  only  a  simple  value,  representing 
two  units ;  but  when  we  write  it  to  the  left  of  another  fig- 
ure or  figures,  thus,  23  or  241,  it  has  a  local  value  as  well 
as  a  simple  value.     This  local  value  depends  on  the  scale  or 
system  of  numbers  employed  and  its  location  in  the  scale. 

• 

15.  Order  of  Figures — The  successive  places  occu- 
pied by  figures  are  often  called  orders.     Thus  a  figure  in 
the  first  place  is  called  a  figure  of  \\\Q  first  order,  or  of  the 
order  of  units  ;  a  figure  in  the  second  place  is  a  figure  of  the 
second  order,  or  of  the  order  of  tens ;  in  the  third  place, 
of  the  third  order,  or  of  the  order  of  hundreds  ;  and  so  on, 
each  figure  next  to  the  left  belonging  to  a  distinct  order, 
the  unit  of  which  is  tenfold  the  size  or  value  of  a  unit  of 


Definitions. 

the  order  of  the  figure  on  its  right ;  and  this  increase  in 
value  from  right  to  left  Ly  ten  constitutes  the  Decimal  Scale 
or  fystem  of  numbers. 

16.  Notation  is  a  method  of  writing  numbers.     There 
are  two  systems,  the  Arabic  and  Roman. 

By  the  Arabic  Notation  numbers  are  expressed  or  writ- 
ten by  Jipures. 

By  the  Roman  Notation  numbers  are  expressed  or  writ- 
ten in  letters. 

pi 

17.  Numeration   is   the  method  of  reading   written 
numbers. 

There  are  two  systems  of  numerating  or  reading  numbers, 
the  French  and  the  English. 

The  French  system  is  the  one  in  general  use  in  the 
United  States  and  the  Continent  of  Europe. 

The  Enf/1Ixh  system  is  that  generally  used  in  England 
and  the  English  Provinces. 


Arithmetical  Examples  and  Exercises. 


FRENCH    S  VST  KM    <  •  K    M   MKKATIoN. 


18.  The  Fivnch  system  separates  liirure^  into  Croups  or 
periods  of  three  figuivs  each.  and  -ives  a  different  name  to 
each  period,  thus  : 


'     2, 
=   7  f 
|J|J 

"  '  ft  v 


i    - 

- 

"  '  ft  I   4^ 

-=        ,  " 


?   ,  r  ic  Bundreda  of 

|ft|J   cc  Tens  of  Octillions. 
(  *  Octillions 

|  "ii    Hundreds  of  ! 
f-f  !'•]    ~    Tens  of 
"     2,  I  J-*   Se|»lilli«)iis. 
I  hmdivd-  of 
Tt-us  of  SrxtilliniH. 

/       >rXtiHi<MH. 

^    Hundreds  of  (t)uintilli<ins. 
^-•i-i  ~'  Tens  of  Quintillionfl, 

i  (   ;-    <v)uintil lions. 
3:^5  /    ~-    Hundreds  <.f  (Quadrillions. 
Ti-n.-  of  Quadrillions. 
Quadrillions. 

Hundn-d--  « 

Tens  of  ' 

Trillions. 

Hundreds  of  Uillions. 

Tens  of  Billions. 

Billions. 

5   -  |  tc  Hundreds  of  Millions. 
|^|-j   o  Tens  of  Millions. 
7    -  (  &  Millions. 
.&  ^  (  t_i  Hundreds  of  Thousands. 
Ifg-j   ?o  Tens  of  Thousands. 
! '  ft  I  j^-  Thousands. 

Hundreds. 

Tens. 

Units, 

The  periods    aliove  Octillions,  in  regular  oulf-r,  are  !u>nilliun>,  Decillions 
I'ndecillions,    Duodecillions,   Tredecillions,   Quatuordecillioiis,    Qnindecilliona 

SexdeciUions,  Se|>t^ndecillions,  Octod'.'cilli'.»!is  >'«tvfuidecil lions   Vii:;intillious, 
&c. 


English  System  of  Numeration. 


ENGLISH    SYSTEM    OF    NUMERATION. 

19.  The  English  system  of  numeration  separates  the 
figures  into  groups  or  periods  of  six  figures  each,  and  desig- 
nates each  period  by  a  distinct  name,  thus : 


ic  Hundreds  of  Thousands  of  Quadrillions. 

GO  Tens  of  Thousands  of  Quadrillions. 

4-*  Thousands  of  Quadrillions. 

^i  Hundreds  of  Quadrillions. 

cs  Tens  of  Quadrillions. 

^  Quadrillions. 

.:  Hundreds  of  Thousands  of  Trillion-. 

i~  Tens  of  Thousands  of  Trillions. 

-j.  Thousands  of  Trillions. 

—  Hundreds  of  Trillions. 
- 1  Ten-  nf  Trillions. 

j£>  Trillions. 

-  Hundreds  of  Thousands  of  Billion-. 

o  Tens  of  Thousands  of  Millions. 

4-  Thousands  of  Billions. 

•-  I  lundreds  of  Billions. 

-:  Tens  of  Billions. 

t~  I  »i  II  ion-. 

4-  Hundreds  of  Thousands  of  Millions. 

/  Tt-ns  of  Thousands  of  Millions. 

-i  Thousand^  of  Millions, 

tc  1  lundreds  of  Millions. 

~-  Tens  of  Millions. 

:;  Millions. 

—  Hundreds  of  Thousands. 
co  Tens  of  Thousands. 

r>  Thousands. 

^.  Hundreds. 

C5  Tens. 

/.  I'n its. 

By  examining  and  comparing  the  two  systems,  it  will  be 
observed  that  they  are  the  same  to  the  ninth  figure  or  the 
hundreds  of  millions,  but  at  that  figure  a  variation  is  made. 
Hence,  if  we  wish  to  know  the  value  of  numbers  higher 
than  hundreds  of  millions,  when  we  hear  them  spoken  or 
see  them  in  print,  we  must  know  whether  they  are  ex- 
pressed according  to  the  French  or  the  English  system  of 
numeration, 


Arithuwtical  K.\ari^cs  and  Examples. 


TIIK  KO.MAN  SYSTFM   OF  NOTATION. 

*JO.      hi  tin-  Roman  system  of  notation  the  letter  1  repre- 
sents '///.•;    Vjfive;    X.  //•//;     L.  //////;    (',  nut-  Jntmlnd;    1>, 
///v  hundred  &ud  M.  <////   ///////>•» ///J.      The  intermediate  and 
'lini:   nnml».-is  are  express. -d   aeeordinL:  to  (lie  Ibllow- 
iiiLT  principles  : 

First. —  Kvery  time  a  letter  i.-  repeated,  its  value  is  re- 
peat-d  :  thu>  II  ivpivseiits  fteo;  XX  rej»r«->ents  ///v uti/. 

Srcond.  —  Wlieiia  Irt ter  o{'  /•  xs/  /•  value  is  jilaeed  before  one 
of  i/rt-utt  r  value,  the  lesser  i>  taken  iroin  the  greater  :  ifplaeed 
afii-r  the  greater,  it  is  to  lie  added  to  it.  Thus,  IV  repre- 
sents f<»'i\  while  V  I  represent.-  >•/./•;  XL  represents  J»rtij^ 
I A  represent.^  >•/./•///. 

Third. — A  line   or  Kir  — .  plaeed  over  a  letter,  increase> 
it>  value   a  thou>and   /inn*.      Thus  X   re]»rt'-ent>   t>  n 
sitnd  ;    L  re]>resents  ////// 


OF   KO.MAN    CHARACTERS. 


I 

II 

111 

IV 

V 

VI 

VI I 

II 11 
IX 

X 

XI 

XII 

XIII 

XIV 

XV 

XVI 

XVII 

XVIII 

XIX 

XX 

XXI 

XXII 

XXIII 

XXIV 


one. 

XXV 

twenty-live. 

two. 

XXVI 

twenty-six. 

three. 

XXVII 

lweuty->even. 

four. 

XXVIIJ 

twenty-ei.uht. 

five. 

X  X  1  X 

twenty-nine. 

->ix. 

X  X  X 

thirty. 

seven. 

XL 

forty. 

eight. 

L 

fifty. 

nine. 

LX 

sixty. 

ten. 

LXX 

seventy. 

eleven. 

LXXX 

eighty. 

twelve. 

xc 

ninety. 

thirteen, 

(• 

one  hundred. 

fourteen. 

cc 

two  hundred. 

fifteen. 

(  «  < 

three  hundred. 

sixteen. 

(  •(  (  •(  • 

four  hundred. 

seventeen. 

I) 

live  hundred. 

eighteen. 

DC 

six  hundred. 

nineteen. 

DCC 

seven  hundred. 

twenty. 

IK.'v  (' 

ei<;lil  hundred. 

twenty-one. 

pcccc 

nine  hundred. 

twenty-two. 
twenty-three. 

M 
MM 

one  thousand. 

two  thousand. 

twenty-four 

MixrrLXXVI 

1876, 

Exercises  in  Notation  and  Numeration.  9 

EXERCISES    IN    NOTATION    AND    NUMERATION. 

21.  In    Writing  Numbers   begin   at  the   left   hand 
with  the  highest   order  and   write  each   period  in   regular 
order,  separating  them  by  com  mas. 

Write  in  figures  the  following  numbers  and  nuni'Titc 
them  according  to  the  French  .system  of  numeration. 

1.  One  thousand,  six  hundred  and  ninety- four. 

2.  Eighteen  hundred  and  seventy-seven. 
.'{.      Twenty-four  hundred  and  six. 

4.  Three  hundred  forty-one  thousand  and  twenty  two. 

5.  Sixty-five  million,  one  hundred  thirty  two  thousand. 
three  hundred  and  eighty -seven. 

6.  Twelve  billion,  sixteen  million,  forty-throe  thousand. 
one  hundred  and  eleven. 

7.  Nine  hundred  thousand,  three  hundred  and  iifiy. 

8.  Six  million,  one  hundred  and  sixty-nine   thousand, 
four  hundred  and  thirty-seven. 

1).  Seventy-six  million,  four  hundred  thousand,  one 
hundred. 

10.  Twenty-two    billion,    one    hundred    three    million, 
five  hundred  seventy-six  thousand,  one  hundred  and  two. 

11.  One  hundred  two  trillion,  one  hundred  twenty  live 
million,  four  hundred  and  three. 

12.  Eight  trillion,  seven  billion  and  seventy-six. 

22.  Write  in  figures  the  following   numbers  and  nume- 
rate them  according  to  the  English  system  of  numeration: 

1.  Four  hundred   twenty-three   thousand,  five   hundred 
and  fourteen. 

2.  Six  hundred   nineteen    thousand,  one  hundred  lil'ly- 
two  million,  t\venfy-one  thousand  and  fortv  seven. 

!•>.  Fifty-three  billion,  two  hundred  twelve  tlmus-nd. 
twenty-six  million,  seventy-live  thousand  three  hundred 
and  eighty-four. 

2:J.  Write  in  the  Koman  System  <;f  Notation  the  follow- 
ing numbers : 

l<->,  12.  14,37,49,  s:1,.  His,  ;,i!>.  i:,i«j.  11701.  ssivrr,. 
13140363. 


1<>          Arithmetical  Exercises  and  Examples. 


A  D  hi  T  I  o\. 

-I.  Addition -"/'"'/•'"*''"// —is  the  prm-ess  i»f  uniting 
two  or  mure  numbers  oi'  the  same  name  or  kind,  so  us  t<> 

make   one   equivalent    IIUlllh'T. 

-").      The  number   obtained   }>\  tliis  process  is  called  the 

Sum  "i  Amount. 

L'O.      The  SiiCIl    Of  Addition    is  a  perpendicular  CFO8S, 

.  called  plus:    it  means  more- ;    thus  7   -\     !'  is  read.  7  plus 

!>,  and    means   that  7  ami  !>  ar  •  to  he    added.      When    used 

after  a  niuiiher.   thus.  5  --)- -.  which  is  read  .">  plus,   it  means 

.">  and  a  small  exo 

'11.  The  SilTll  Ol'  Kqimlity  is  •  ^t  is  read  equals, 
or  eijual  to.  and  denotes  that  the  numhers  hetween  which 
it  is  placed  are  njual  to  each  other  ;  thus  7  |  !>  ^=  1C 
means  that  7  and  !>  added  art1  njual  to  Hi.  The  expression 
is  read.  7  ]>lus  !>  equals  1  <i. 

A  Numerical  Equation  is  an  equality  lu-tu-een 

t  \vo  numerical  expressions,  which  tliou^h  ditleriiiL:;  in  form 
from  each  other  are  equivalent.  Macli  expression  is  called  a 
term  of  the  equation.  Thus  5  +  8  =  13  is  a  numerical 
equation  in  which  the  5  ;  S  is  called  the  first  memhei1  of 
the  equation  and  13  the.  see-md  meinher.  and  hoth  are  called 
the  terms  of  the  equation. 

L>(J.  rrhn-ijtli'  <>f'  Addition.  Xumhers  of  the  same  kind, 
order  or  character  only,  can  he  added.  Thus  we  cannot  add 
-  apples  and  3  oranges,  nor  5  pounds  of  sui^ar  and  (I  boxes 
oi'  peaches  ;  nor  (I  units  and  f>  hundreds;  nor  2  and  •}.  etc. 
\Ve  can  only  add  apples  to  apples,  oranges  to  oranges,  suirar 
to  suu'ar,  peaches  to  peaches,  units  to  units,  hundreds  to 
hundreds,  halves  to  halves,  fourths  to  fourths,  etc.  We 
can  enlist  together  things  of  different  kinds,  apples,  peaches, 
fiances,  etc..  hut  by  coll-ectitii:  them  together  we  do  not 
increase  the  number  or  sum  of  either  aud  hcuce  there  is 
no  addition. 


Addition   Tables.  1 1 


ADDITION    TABLE. 

oU.     No.   I. 

Kan.     In  h-iiniiiii:-  thr.-M-  tal.l.-s  ami   hamllin.ir  all   numliiTs.  all   int<Tiiir«liat" 
\\..n.l- ;l|Mi   thoughts  that  occur  l,,-t\v.-cii    th..    iitmil"-r-  to  I,,-  .-..mliim-il  ami  the 

iv-nlt  of  tin-  d«-iiv.l  cuml, inatioii  >lh,ul.|  I mitt.'.l.     Thus  in.-t.-a.l  ..f  savini: 

or  thinking,  that  ~1  ami  -  ar.-  1.  :'•  an-1  5  ura  >.  etc.,  M$  Ot  tliinU   1  :   v  :   etc. 

/•.\rj>/<(itn(i<jH — In    this    t;ible 

1  .,  we  show  20  different  combioations 

1 

ol'  (he  (J  signitieant  ii-iuvs  to  pro- 

2  duce    results    iroin    1    to    (J.      It 
2  -^                           may  be  said  that  three   lv   make 

9  '-i 

o,  three  2's  make  (J,  etc.,  and  that 

5  they  are  regular  combinations;  but 

4.3 


we  see  by  the  table  that  two  l's 

1.2.:; 

~}.  I.;;  are  2,  and  that  two  IT  are  1,  ete. 


\  -2  ;>>  Hence,  though  the  table  does  not 

6.5.4         7 

contain  all  the  possible  combma- 

1       4>     O       1 

._'""J'         8  tions,  it  does  contain  all  that  are 

i  .0.0.4 

essential  and  of  value  in  this  con- 

1.2.3.4  g 

8.7.6.5  nection. 


Arithmetical  £\rni;.srs  un.l  Examples. 
ADDITION    TAB  LK. 


]•_>;;   j  ;, 

:i  x •- j;  -}  Explanation.  —  In  this  table  we 

>lin\v  ihi-  LT)  (litlt'ivnt  combinations 
L'.:5.  I. :> 

:>. S.T.I;  of  the  :>  significant   ii^mvs   th<> 


.>',  "  (.  sum  of  which  equals  f> •//  or  more. 

1'2 
'•s-<(»-  '!«>    attain    rapidity   in    adding,    it 

I  ;,.i;  is  absolutely    necessary    that,  the 

MS" 

'_^__  learner  should  be  so  familiar  with 

•">."'. 7  ,  .        these    eombinatiuns    that    he    can 

9.8.7 

instantly    see  the   result    without 

15       adding,    i.  e.  he  must    know  the 

J.a 

result  by  the  combination,  just  as 

r+  o 

y'g  l^       he  knows  the  value  of  4,  or  5,  by 

the  combination  of  lines  forming 

8  -j  *r 

9  the  figure,  or  as  he  knows  the  pro- 

nunciation   of    a    word    without 

18 

spelling  it, 


The  rapid  increasing  and  decreasing  operations  in 
the  science  of  numbers  depend  upon  the  capacity  of  the 
calculator  to  instantly  apprehend  and  accurately  apply,  the 
result  of  two  or  more  figures,  no  matter  how  they  may 
be  combined.  And  the  object  of  these  tables  is  to  aid  in 
acquiring  the  desired  capacity* 


Addition  and  Subtraction  Tables* 


13 


K  w 

H  ^ 

«  .  « 

EC  H 


rH  <M  CO 


rH  ^1  CO 


L-  t-  t-.  t—  t»  t- 


rH  -M  CO  -t  10  CO 


00  GO  00  00  GO  GO  00 


rH  £1  CO  ^  »O  CD 


•S.§ 


0 

M 

H 


c  ^  o 

*  1  S 

O  03  C 

°  "**  5 

§£"S 


C  &^ 

S  o-° 

o  s  .. 

o  °°  c 

10  «  .2 


.a  *-  ^ 


.2 


14          Arithmetical  Exercises  and  Examples. 


ADDITION   AND  StT,TI!A<  TK>N   TAI'.LKS. 
33.     TA15LK   IV. 


1  A 

i  7=100 

26  A 

?=100 

:.i  , 

loo 

76  & 

?=lfO 

1 

100 

27 

100 

52 

loo   77 

loo 

3 

loo 

28 

100 

53 

loo   78 

100 

4 

100 

29 

100 

54 

loo 

100 

5 

100 

30 

100 

loo 

100 

6 

100 

31 

100 

56 

loo 

si 

100 

7 

100 

32 

100 

57 

100 

100 

8 

100 

33 

100 

58 

100 

83 

100 

!> 

100 

34 

loo 

r>9 

loo 

100 

10 

100 

100 

100 

100 

11 

100 

36 

100 

til 

100 

100 

12 

100 

37 

loo 

62 

100 

87 

100 

13 

100 

38 

100 

63 

100 

88 

100 

14 

100 

39 

100 

64 

100 

89 

100 

15 

100 

40 

100 

65 

100 

90 

100 

16 

100 

41 

100 

66 

100 

91 

100 

17 

100 

42 

10* 

67 

100 

92 

loo 

18 

100 

43 

100 

68 

100 

93 

100 

19 

100 

44 

100 

69 

100 

94 

100 

20 

100 

45 

100 

70 

100 

95 

100 

21 

100 

46 

100 

71 

100 

96 

100 

22 

100 

47 

100 

72 

100 

97 

100 

23 

100 

48 

100 

73 

100 

<J8 

100 

24 

100 

49 

100 

74 

100 

99 

100 

25 

100 

50 

100 

75 

100 

We  present  this  table  to  aid  the  learner  in  instantly  seeing 
the  difference  between  100  and  any  number  from  1  to  99.  It 
is  of  special  value  in  addition  and  subtraction,  and  all  who 
expect  to  become  rapid  Calculators,  must  be  proficient  in  this 
character  of  work. 


Signs  and  Abbreviations.  15 


SIGNS  AND  ABBREVIATIONS. 

84.     The  following  are  the  principal  signs  and  abbrevia- 
tions in  general  use  among  merchants  and  business  men  : 

(a    At.  To.  Company. 

'•'/,.  Account.  (1r.   Credit  or  creditor. 

I1  Oue  and  one-quarter.         J>r.   Debit  or  Debitor. 
la  One  and  one-half.  Gal.   Gallons. 

]<H  One  and  three-quarters.     Ps.   Pieces. 
Per.  Yd.    Yards. 

Pound  (weight).  Fr't.    Freight. 

$  Dollar  or  dolhns.  ller'd.   Keceived. 

/  Cent,  or  cents.  Pay't.   Payment 

%    Per  cent,  or  per  centum.  lust.  This  month. 
A  int.  Amount.  Prox.  The  next  month. 

Blil.    Barrel.  VI.  The  last  month. 

Doz.    Dnzrn.  U    Pound  Sterling. 

B.  L.    Bill  «.f  Lading.  0.  K.   All  Right. 

Blk.    Black.  Fr.   Franc,  French  coin. 

Shipt.  Shipment.  Fwd.   Forward. 

Sunds.   Sundries.  Bal.   Balance. 

Dft.  Draft,  Cons't.  Consignment. 

Com.   Commissioo.  hhds,  hogsheads. 

Do.  The  same.  Mdse.   Merchandise. 

/  Shillings,  thus  2/6  two  shillings  and  sixpence. 
iMk.   Marks,  the  German  monetary  unit. 
I      Check  mark,  correct,  approved. 

0    Cifrao,  used  to  separate  the   milreis  from  the  rcis  in 
Brazil  money. 


17  doz.  §,-*,.  S/\.  §T7T=17  doz.,  4  of  which  are  at  810 
per  doz.,  G  (<y  $12  and  7  @  $15. 

8  doz.   |  @  5  /  f  (a>  I,  2  doz.   No.  4  (oj  5  shillings  per 
doz.,  and  (5  doz.  No.  5  (fV  4  shillings  sixpence  per  dozen. 


16          Arithmetical  Exercises  and  Examples. 

35.     Name  the  unit  result  of  the  following  numbers : 
!)      898989887       9       5       6 
9866442298797 


3 

8 

7 

6 

8       5 

5 

6 

9 

4 

7 

5 

9 

3 

3 

4 

4       7 

5 

(i 

1 

7 

7 

8 

3 

4 

5 

1 

1 

9 

8 

g 

3 

6 

5 

5 

2 

2 

3 

3 

2 

1 

5 

7 

4 

7 

7 

9 

8 

7 

8 

2 

4 

6 

8 

9 

3 

7 

1 

8 

4 

6 

5 

4 

9 

7 

9 

7 

4 

8 

2 

7 

7 

8 

5 

6 

8 

8 

5 

5 

8 

9 

9 

8 

9 

8 

8 

5 

6 

9 

For  further  explanation  of  Addition,  the  importance  of 
it,  and  the  most  rapid  processes  of  adding  see  Soule's  Con- 
tractions in  Numbers. 

EXERCISES. 

1.  Write  all  the  combinations  of  two  figures  that  make 
10,  11,  12,  13,  14,  15,  16,  17  and  18. 

2.  Commence  with  1  and  orally  add  thereto  2,  and  con- 
tinue to  add  2  to  the  successively  occurring  sums  until  you 
produce  21.     Thus  3,  5,  7,  9,  11,  13,  etc. 

3.  Commence  with  1  and  in  like  manner  add  3  until 
you  produce  31.     Thus  4,  7,  10,  13,  etc. 

4.  Commence  with  1  and  in  like  manner  add  4  until 
you  produce  41. 

5.  Commence  with   1  and  in  like  manner  add  5   until 
you  produce  51. 

6.  Commence  with  1  and  in  like  manner  add  6  until 
you  produce  61. 


Examples  in  Addition.  17 

7.  Commence  with  1  and  in  like  manner  add  7  until 
you  produce  71. 

8.  Commence  with  1  and  in  like  manner  add  8  until 
you  produce  81. 

9.  Commence  with  1  and  in  like  manner  add  9   until 
you  produce  91. 

10.  Oially  add  by  2''  until  you  produce  20. 

11.  «  "  3'*  "  "  30. 

12.  "  "  -1  v  "  "  40. 

13.  "  "  ")  "  "  50. 

14.  "  "  6'8  "  "  GO. 

15.  "       7'8         "  "         70. 

16.  "         "       8'8         "  "         80. 

17.  t{         "        9"          "  "         90. 

18.  "         "      10'"          "  k'       100. 

19.  Commence  at  1  and  orally  add  by  3  and  5  altern- 
ately until  you  produce  100. 

20.  Commence  at  1  and  orally  add  by  4  and  7  altern- 
ately until  you  produce  100. 

EXAMPLES   IN    ADDITION. 

36.     Add  the  following  numbers  :  6376,  564,  309,  485 
and  5092. 

OPERATION. 

w  j  Explanation. — In    all    addition    pro- 

||  blems   we   firgt  write   the  numbers  so 

|lc|  that  units  of  the  same  order  will  stand 

in  the  same  column,  i.  e.,  units  in  the 
6o7o  units  or  first  column;  tens  in  the  tens 

r  f>(H  or   second   column;    hundreds   in   the 

;;OJI  hundreds  or  third  column  and  so  on 

through  the  numbers.  We  then  begin 
at  the  units  or  first  column  and  add 
the  columns  separately.  In  adding  the 
first  column,  we  commence  with  the  2 
Swn  12,826  and  5,  and  name  only  the  successive 

1^  results  thus,  7,  16,  20,  26;  which   is  2 

tens  and  6  units;  the  6  we  write  in  the  first  place  or  column  of 
units  and  place  the  2  tens  which  is  to  be  carried  to  the  column  of 
tens  directly  below  the  6  in  a  small  figure.  Then  adding  the  2 


18          Arithmetical  Exercises  and  Examples. 

tens  to  the  tens  column,  we  say,  11,  10,  !!:>,  .".2;  which  is:*  hundred 
and  2  tens:  the  2  tens  we  write  in  the  column  of  tens  and  place 
the  3  hundreds,  which  is  to  be  carried  to  the  hundreds  column 
directly  under  it.  Then  adding  the  3  hundred  to  the  hundreds 
column,  we  say,  7,  10,  15,  18,  which  is  1  thnumtml  and  S  hun- 
dred; the  8  hundred  we  write  in  th«  hundreds  column  and  the 
carrying  figure,  1  thousand,  directly  under.  Then  adding  ihn 
1  thousand,  to  the  fourth  or  thousands'  column,  we  say  <;,  \'2, 
which  is  I  (en  thousand  and  2  thnuxtinrl,  and  this  being  the  last 
column  to  add  we  write  the  figures  in  their  respective  columns 
and  produce  12826  as  the  siim  of  all  the  numbers. 

When  adding,  set  the  result  in  pencil  figures,  being  careful 
to  place  the  carrying  figure  or  figures  directly  beneath  the  unit 
figure  of  each  column  added  as  shown  in  the  preceding  problem. 

PROOF  OF  ADDITION. 

The  best  proof  of  the  correctness  of  addition  is  to  be 
proficient  in  your  work,  and  then  re-add  the  columns  in 
the  reverse  direction. 

What  is  the  sums  of  the  following  groups  of  numbers  ? 

(3)  W  (5)  (6) 

780  89  777  9040 

1261  706  888  1288 

537  73  999  9907 

309  4009  666  6543 

6987  8888  645  2018 


Examples  in  Addition.  19 


Add  the  following  groups  of  numbers  : 


818 

(8) 

412 

582 

(10) 

328 

(ii) 
809 

(12) 

981 

390 

297 

578 

346 

523 

350 

970 

318 

757 

386 

605 

269 

270 

824 

420 

672 

848 

789 

752 

932 

731 

793 

945 

696 

843 

373 

542 

£64 

397 

136 

805 

570 

853 

905 

684 

169 

129 

876 

684 

448 

976 

295 

768 

444 

743 

404 

666 

468 

9U4 

102 

915 

151 

217 

687 

972 

814 

080 

148 

879 

825 

114 

331 

637 

263 

516 

951 

340 

554 

917 

295 

259 

784 

545 

101 

650 

101 

890 

122 

022 

m 

411 

401 

864 

440 

749 

490 

237 

874 

565 

450 

717 

876 

349 

898 

150 

414 

222 

902 

489 

769 

514 

654 

234 

390 

698 

243 

446 

789 

100 

484 

228 

174 

576 

458 

305 

235 

433 

952 

489 

747 

272 

380 

949 

683 

394 

030 

729 

.624 

087 

574 

407 

241 

955 

897 

702 

956 

812 

477 

177 

477 

849 

658 

798 

081 

20          Arithmetical  Examples  and  Exercises. 


Add  the  following  groups  of  numbers: 


(13) 

864 

,H) 

B77 

595 

(16) 

849 

(17) 

539 

257 

363 

305 

249 

:)>77 

476 

629 

420 

n;.; 

027 

702 

426 

145 

982 

830 

651 

235 

684 

174 

217 

221 

543 

492 

144 

326 

232 

502 

950 

343 

176 

111 

151 

113 

446 

002 

767 

871 

:;^7 

438 

834 

182 

644 

512 

516 

455 

540 

955 

747 

814 

247 

328 

919 

156 

376 

331 

633 

358 

989 

106 

468 

281 

(524 

149 

855 

872 

189 

S2S 

581 

268 

954 

694 

177 

986 

491 

002 

126 

788 

885 

817 

888 

693 

136 

866 

264 

918 

992 

682 

564 

044 

294 

IN!) 

202 

355 

163 

922 

896 

259 

548 

223 

764 

116 

597 

365 

521 

921 

911 

814 

329 

208 

530 

515 

866 

277 

678 

662 

874 

735 

179 

476 

040 

704 

528 

393 

129 

716 

821 

387 

584 

550 

659 

778 

802 
584 

457 

587 

848 
255 

025 
202 

888 
932 

Examples  in  Addition.  21 


Add  the  following  groups  of  numbers: 


19 

20 

21 

22 

23 

24 

883 

792 

743 

153 

919 

547 

356 

414 

560 

214 

620 

380 

595 

454 

871 

248 

922 

616 

638 

366 

349 

636 

369 

874 

679 

464 

955 

549 

158 

682 

594 

933 

936 

f>!>4 

862 

232 

953 

686 

746 

783 

874 

713 

178 

641 

793 

225 

935 

499 

215 

939 

798 

61!) 

951 

874 

119 

201 

324 

232 

959 

779 

753 

S71 

687 

478 

865 

622 

311 

438 

843 

484 

724 

718 

1  S2 

218 

421 

252 

645 

180 

686 

869 

586 

648 

148 

477 

896 

189 

518 

551 

227 

396 

996 

595 

959 

995 

193 

495 

293 

521 

152 

475 

947 

568 

2(52 

727 

572 

D77 

797 

130 

515 

259 

425 

362 

736 

111 

833 

585 

458 

485 

328 

682 

745 

177 

217 

631 

803 

685 

125 

413 

841 

194 

729 

996 

245 

825 

972 

698 

169 

492 

968 

868 

489 

258 

845 

194 

540 

799 

386 

277 

477 

864 

Arithmetical  Exercises  and  Example*. 


-11  28 

15656:;  13331H2 


IMJ4SJH'  -.VJ7  8368G9               7391  573 

f>7sr,7*  78754d  2.§M24<>               35175(11) 

5775!>4  !U  14:12  7H5183               8598674 

GGSU7S  wwi<  :;i5!>^7 

6<;<M;:>7  :»7s:;iji  »;.viG78 

5398  678789  I5(;.i:^ 

(;c,47:)«;  :H;H;T:{  :ur>7l8 

795568  895437  7»;:>391 

(IDDC.sn  569128  \\r.\\X\              7893344 

i;s!i7si;  i;7snsi>  137987 

688968  ><;'J771  r)i;«;7-:» 

i):;r,7  668339  :>4  1321 

7788JHJ  'jrHiL'.-U  891389 


431348  4235564 


L>1>.  Add  6,  s.  <>,  7,  6,  8,  r>,  4,  95  4,  8,  7,  6,  9,  14,  19, 
IS,  ITT,  HS.  47,  59,  65,  74,  83,  9i'.  Ans.  632. 

30.  Add  528,791,  14389,888,91361,587,301,7004, 
52800.  71  <H5.  42S81.  Am.  218,636. 

31.  Add  476010,  51873,  98,- 48932,  3581427,  67843, 
21050,  3672.  Ans.  4,250,905. 

82.  Add  63,  94,  85,  74,  63,  52,  41 ,  3<»,  48,  57,  66,  75, 
84,  93,  27,  18,  60,  80,  19,  88,  99,  77,  66,  55,  44,  33,  22, 
11,  98,  97,  !>6,  Sii,  76,  65,  54,  43.  Ans.  2248. 

33.  Add  seven  million  four  thousand   and   ninety-six, 
and  three  hundred  eighty-seven  thousand  five  hundred  and 
sixty- tvo.  Ans.  7391658. 

34.  Find    the   sum    of    4888765,    92238,    1600084, 
8888888,  99999999999,  4100000808707  and  222222333- 
333444444.  Ans.  222226533349723125. 

35.  Find  the  sum  of  999999999,  88888888,  7777777, 
666666,  55555,  4444,  333,  22,  1,  and  sixty-three  millions. 

Ans.  1160393685. 


Examples  in  Addition.  23 

80.  Add  781),  67!).  <J87,  140018,  11)1070,  871230432, 
4!)70G,  40000,  80000000  and  eleven  hundred  and  eleven. 

Ans.  951,654,71)2. 

37.  Add  five  hundred  thousand  nine  hundred  thirty- 
nine,  and  eleven  thousand  eleven  hundred  and  eleven. 

Ans.  513050. 

37.  Dollar  and  (-cut  si^ns.    The  dollar  sign  is  $, 
and  the  cent  sign  is  c.      When   the  dollar  sign  is  placed 
before  numbers  they  are  read  as  dollars.     Thus  $45  is  read 
45  dollars.     When  the  cent  sign  is  placed  after  numbers 
they  are  read  as  cents.     Thus  14^  is  read  14  cents.     When 
dollars  and  cents  are  written  together  the  cents  are  separated 
from   the  dollars  by  a  point   (.)   and  the  sign  of  cents  is 
omitted.     Thus  $10. 45  is   read  1(5  dollars  and  45  cents. 

Since  there  are  100  cents  in  1  dollar,  cents  always  occupy 
two  places  and  only  two  in  connection  with  dollars.  When 
the  number  of  cents  is  less  than  10  a  mnif/lit  must  be  used 
to  iill  the  tens  column  or  the  first  place  at  the  right  of  the 
point.  Thus  8  dollais  and  5  cents  are  written  $8.05. 

When  cents  only  are  written  they  are  expressed  as  fol- 
lows :  25  cents,  or  25/  or  $.2."). 

When  writing  numbers  representing  dollars  and  cents  for 
the  purpose  of  addition,  they  must  be  set  so  that  dollars 
will  be  under  dollars  and  cents  under  cents  in  the  regular 
order  of  units,  tens,  hundreds,  etc.,  and  the  points  (.)  that 
separate  dollars  and  cents  must  be  in  a  vertical  line. 

The  dollar  sign  ($)  and  the  point  (.)  should  never  be 
omitted  when  writing  dollars  and  cents. 

38.  Add  $14.50     $34.  Ki     *75.         $     .88     $180.40 

8.  D.OS          !.45       11.  48.08 

4.25       14.83       <»7.0(i         5.13         91.16 

12.15         8.  .35         7.02  7.05 


S21.03     $326.G!) 


24          Arithmetical  Exercises  and  Examples. 

Add  $821.         8521.16     8     1U5  $431,       $194.15 

040. SO         88.2.")         80.  124.             8.05 

9.13       19.30        17.  381.         73.75 

75.20           8.                 .65  569.             0.13 

100. 1C)            4.07            6.10  S27.                .95 


$1146.18     $635.78     SI  13.20      82332.       8283.03 

48.     Add    $8.12,   $1),  $.50,  $3.40,  $37.05,  $.75  and 
$12.12.  Ans.  870.D4. 

4!).     Add  8  i:5. 10,  $17.  $5,  48C,  75/,  $11,  $24.14,  $3. 

Ans.  $104,17. 

50.  Add  $108,  $97.16,  881.12,  $.75,  $8,  $6.40,  25/, 
$18.  Ans.  $322.68. 

51.  Add  $580.10,  $671.23,  87!)  1.1)8,  $88,  45/,  5/, 
$3.10.  Ans.  $2,137.91. 

52.  Add  §999.99.  $sss.S8,  8777.77,  $<><>«;.ii<;.  8555.55, 
84-14.44,  $333.33,  $222.22,  $111,11  and  I/. 

Ans.   8-4. 999. 96. 

53.  Add  $1)87.65.  $876.54.  8765.4:;,  8(554  32,  8543.- 
21,  $123.45,  $234.56,  $345.67,  $456.78,  $567.89,  $678.90 
and  $789.  Any.  $7,123.40. 

54.  Middle-miss  bought  a  hat  for  $2,  a  coat  for  $9.50, 
a  pair  of  shoes  for  $2.75,  a  pair  of  pants  for  $4,  a  vest  for 
$1.75,  and  had  $41.05  left.     How  much  money  had  he  at 
first?  Ans.  $61.05. 

55.  Miss  Smith  paid  for  a  broom  35/,  for  soap  $1.60, 
for  starch  75/,  for  matches  5^,   for  salt  15/,  for  sugar 
$1.50,  for  rice  $2,  for  butter  80/,  Graham  flour  $1.25  and 
for  a  hygienic  cook  book  $1.     What  was  the  sum  paid  for 
all?  Ans.  $9.45. 

56.  Prophet  paid  for  a  reader  $1.35,  for  an  arithmetic 
$1.50,  for  a  history  $2,' for  a  set  of  drawing  instruments 
$3.70,  for  paper  $.60,  for  pens  $.15,   for  ink  $.05,  for  a 
pair  of  Indian  clubs  $3.50,  and  for  the  boy's  own  book  $1. 
What  did  all  cost  ?  Ans.  $13.85. 

57.  Conrad  paid  $1.75  for  Chesterfield's  letters  ;  $1.80 
for  Cutter's  Anatomy,  Physiology  and   Hygiene ;    $1.75 


^Examples  in  Addition.  25 

for  Comb's  Constitution  of  Man  ;  $1-25  for  How  to  Read 
Character  by  Wells;  $1.50  for  .Nordhoff's  Politics  for 
Young  Americans;  $1.75  for  Physical  Perfection  by  Jac- 
ques ;  $4  for  Plutarch's  Lives  ;  $8  for  Shakspeare's  Works ; 
$2  for  the  Literary  Header ;  $6  for  Carey's  Social  Science ; 
$5  for  Parson's  Laws  of  Business;  $5  for  Soule's  Philoso- 
phic Work  on  Commercial  and  Exchange  Calculations,  and 
$1  for  Cushing's  Manual.  How  much  did  he  pay  for  all? 

Ans~  $40.80. 

58.  If  you  should  travel  by  rail  160  miles,  by  steamer 
214,  and  walk  8,  how  far  would  you  travel? 

Ans.  382. 

59.  A    planter   raises    9842    pounds    of   sugar,    2351 
pounds  of  cotton,   1827  pounds  of  rice,  3840  bushels  of 
corn,  325  bushels  of  sweet  potatoes  and   194  bushels  of 
beans.     How  many  pounds  and  how  many  bushels  does  he 
raise  in  all?  Ans.  14020  pounds,  4359  bushels. 

(JO.  Conrad  loaned  to  Purccll  8:>  :  to  ( Iresham  $3.50  ; 
to  Ilanna  75/ ;  to  Mitchell  85;';  to  Sweeney  5/ ;  to 
IJothick  $1  ;  to  Keen  25/J  to  Abbott  75/ ;  to  Prophet 
50/.  What  sum  did  he  loan  to  all?  Ans.  £16.65. 

til.  Keen  has  Si  I3.o:>  ;  Courct  $91  ;  McCoard  $18.30; 
Bush90/;  Nevers  25  c  ;  Fischer  *:>.<>.'>  :  Heck  $9  ;  Meyers 
s<; ;  Levy  $7  :  Brown  $7  ;  Kk-e  $45  ;  Shotwell  £27  ;  Wise 
$15.80  ;  Moffett  $5.50  ;  Limlsey  888.70.  How  much  have 
all?  Ans.  $460.55. 

62.  A  merchant  bought  four  adjacent  lots  of  ground 
for  $6850.     He  built  a  house  thereon   which  cost  $11875. 
Paid  for  fences  $912  ;  for  flagging  $1819.55  ;  for  furniture 
$3481.12.      How  much  did  the  whole  cost? 

Ans.  $24,937.67. 

63.  If  you  pay  $175  for  a  horse,  $450  for  a  carriage, 
$75   for  a  set  of  harness,   s:;s   for  a  saddle  and  bridle  and 
$6.50  for  a  whip.     What  will  the  whole  cost? 

Ans.  $744.50. 

64.  A  planter  has  54  cows,  321  sheep,  174  mules,  23 
horses,  42  oxen,  43  calves,  7  colts.     How  much  live  stock 
has  he  altogether  ?  Ans.  664. 

Q 


-»•          Arithmetical  Exercises  and  Examples. 

!».">.  A  merchant  bought  at  one  time  250  barrels  Flour 
for  81  ")()() ;  at  another  .'>  if)  banvls  for  8-  H5  ;  and  at  another 
21)0  barrels  for  £1625.  How  many  barrels  did  he  buy  and 
what  was  the  total  cost7  Ans.  795  Bbls., 

$5540  Cost. 

66.  The  weight  of  ten  bales  of  cotton  is  as  follows  :  481, 

503,  :-ws,  4r>2,  470.  IT(J,  mi,  :;:»T.  n;:i,  511  pounds,  what 

is  the  total  weight?  Ans.  4565. 

67.  Bought  at  one  time  43  yards  of  calico  and  32  yards 
of  silk  ;  at  another   104  yards  of  calico  and  24  yards  of 
silk,  and  at  another  96  yards  of  calico  and  48  yards  of  silk. 
How  many  yards  of  each  kind  did  I  buy? 

Ans.  Calico  2415,  Silk  104. 

68.  Paid  $425  for  a  lot  of  BUgar,  s  1  20  i;,r  rice  and  §75 
for  potatoes.     Sold  the  sugar  at  a  profit  of  $  H  and  the  rice 
and  potatoes  at  cost.     What  did  I  get  for  the  whole? 

Ans.  £661. 

69.  From  New  Orleans  to  the  Iligolets  is  31   miles  ; 
hence  to   Montgomery,  18;  hence  to  Bay  St.   Louis,  3; 
hence  to  Pass  Christian,  6  ;  hence  to  Mississippi  City,  13  ; 
hence  to  Biloxi,  9  ;  hence  to  Ocean  Springs,  4  ;  hence  to 
East  Pascagoula,   16;  hence  to  St.   Elmo,   21;  hence  to 
Mobile,  20.     How  many  miles  to  Mobile?       Ans.  141. 

70.  From  New  Orleans  to  Kenner  is  10  miles  ;  hence 
to  Manchac,  27  ;  hence  to  Ponchatoula,  11  ;  hence  to  Ham- 
mond, 4;  hence  to  Amite,  16  ;  hence  to  Tangipahoa,  10  ; 
hence  to  Osyka,  10  ;  hence  to  Magnolia,  10  ;  hence  to  Me 
Comb  City,  7;  hence  to  Summit,  3 ;  hence  to  Bogue  Chitto, 
10  ;  hence  to  Brookhaveu,   10  ;  hence  to  Beauregard,  11  ; 
hence  to  Crystal  Springs,  19  ;  hence  to  Terry,  9  ;  hence  to 
Jackson,  15;  hence  to  Madison,  13;  hence  to  Canton,  11. 
How  many  miles  is  it  to  Canton  ?  Ans.  206. 

71.  A  young  man  paid  $125  for  a  year's  tuition  at  col- 
lege, $22.50  for  books,  lost  $40,  and  has  $378.35  on  hand. 
How  much  had  he  at  first?  Ans.  $565.85. 

72.  A  boy  gave  Jane  6  oranges.  Kate  4,  John  3,  he 
ate  2,  and  had  5  remaining.     How  many  had  he  at  first  ? 

Ans.  20. 


Examples  in  Addition.  '11 

73.  Louisiana  contains  41 255  square  miles  ;  Mississippi, 
47156;    Texas,   237504;    Arkansas,    52198;    Tennessee, 
45600;    Kentucky,   37680;    Alabama,    50722;    Georgia, 
52009;  South   Carolina.   293S5  :    North   CJardifta,  50704; 
Missouri,    07380;     Virginia,    r,  1:552:     Maryland.    11124: 
Florida,   592C.8  ;   California.    1KS982.      How   many   square 
miles  in  the  fifteen  states?  Ans.   1,032,319. 

74.  The    population    of    London    is    3311000;     Paris. 
1852000;   Si.  Petersburg,  Iili7000  :    Pun  Janeiro.  2750(10: 
Constantinople.  400000  ;  Vienna.  S3  1-000  ;   Berlin,  S25000  : 
Lisbon,  224000;    IVkin.    ir,!SOOO;   Tokio  or  Jeddo,  790- 
000;     Bomhav.     1117000;     Madrid.     332000;     (ilasgow, 
489000;   Dublin,  31 1000;   Amsterdam.  27SOOO  ;  Brussels, 
176000;  Stockholm.  139000;  Copenhagen,  181000;  Cairo, 
(Egypt)    351000;   Tunis.    125000.      What    is   the  popula- 
tion of  all?  Ans.    13,858,000. 

75.  The  lenizth  of  the  Mississippi  Hiver  is  1200  miles  : 
of  the    Nile,   4000:    Ama/on.   3750  :    Yenisei.  3  400  ;   ()l,i. 
3000:   Yang-tse-Kian-  3320;   tfiger,  3000 ;   Lena,  2700  : 
Amoor,  2<i50;    Yol-a.   2000;   (Jan-vs.    liJOO;    Brahmapoo- 
tra, 2300  .    |j;l    Plata.   2300;    Macken/.ie.    2:;oo  ;   St.    Law- 
rence, 2000;  Saskatchewan.  1900;  OriiHH-o.  1550;  Colum- 
bia,   1020;     Colorado.    (iOO  ;     Yukon.    HJOO;     Kcd    lliver. 
1500.      What  is  the  combined  len-th  of  all  ? 

Ans.  50.090. 

7<>.  Lake  Sii])erior  is  100  miles  in  Iciiiith  .  Lake  Michi- 
gan, 320  ;  Lake  Ihmm,  210;  Lake  El ie,  240  ;  Lake  On- 
tario. 180;  Lake  Baikal,  375:  L-ike  Pontehartrain,  40. 
What  is  the  comhined  length  oi'  all  ?  Ans.  1795. 

77.  There    are    in    the    world    3910(10000    Christians: 
500000000  Buddhists:   1  150000m)  Brahmins;  100000000 
Confucians;    15000000   Slmdoan  :    199000000  Mohamme- 
dans ;   7000000  Isndites.      1  low  many  combined  ? 

Ans.    1.3(iO,000.000. 

78.  Mount   Kvei-est  of  the  Himalaya  chain  in  Asia  the 
highest  point   on    the  globe,  is   29002    feet   high;    Mt.  St. 
E.ias,  the  highest  mountain  in  North  America,  is  17900 
feet;  -Mt.  Illampu,  the  highest  mountain  in  South  America, 


28         Arithmetical  Exercises  and  Examples. 

is  24812  feet;  Mt.  Blanc,  the  highest  mountain  in  Europe, 
is  15780  feet;  Mt.  Kilima  Xj.iro,  the  highest  mountain  in 
Africa,  is  200G5  feet;  Mt.  Koseiusko.  the  highest  mountain 
in  Australia,  is  717U  feet.  What  is  (lie  nunhined  heiirht 
of  all?  Ans.  114.795  feet. 

70.  l>v  the  census  of  1  STO.  the  population  of  New  York 
was  942992;  Philadelphia,  674022;  Brooklyn.  :J!M>niM)  ; 
St.  Louis,  310864  ;  Chicago,  2!»«.»77  ;  Baltimore,  2U7354; 
Boston,  2:>or>2i;  ;  Cincinnati.  2Hi2:!'J:  New  Orleans'. 
19141S:  San  Francisco,  140473  ;  Buffalo,  117714  ;  Wash- 
ington, 100199;  Newark,  105059;  Louisville,  100753; 
Mobile,  32034;  Galveston,  13818;  Memphis,  40226. 
What  is  the  population  of  all  combined  ? 

.  ADS.  4211571)7. 

80.  On  Monday  85482  persons  entered  the  gate -s  at  the 
Centennial  Exhibition,  Philadelphia;   on  Tuesday.  10S121  : 
on  Wednesday,   98792;   on  Thursday.    HI  9515;   on  Friday, 
103819,  and  on  Saturday,  174587.     How  many  entered  in 
the  six  days?  Ans.  (>(;:». ()5 4. 

81.  The  standing  army  of  the  United  States  is  liL'OOO  , 
of  Great  Britain  and  Ireland,  192000  ;  of  France,  454000  ; 
of  the  German  Empire,   402000;  of  Russia.   7<J<iOOO  ;  of 
Spain,  284000;  of  Switzerland,  201000;  of  Italy,  205000  ; 
of  Brazil,  25000  ;  of  Mexico,  21000;  of  Turkish  Empire, 
93000;  of  Sweden,  150000;  of  Holland,  62000;  of  Por- 
tugal,  33000;  of  Belgium,  40000.     How  many  men    in 
all?  Ans.  2,960,000. 

82.  Homer  was  born  733  years  before  the   Christian 
Era.     How  many  years  from  the  birth  of  Homer  to  the 
year  1876?  Ans.  2609. 

83.  During  the  fiscal  year  ending  Sept.  1st  1876,  the 
receipts  of  cotton  at  various  points  were  as  follows  : 

New  Orleans,  1401563  bales;  Galveston,  465529 ;  Mobile, 
371298;  Savannah,  521437;  Charleston,  389698;  Wil- 
mington, 78267;  Norfolk,  469997;  Baltimore,  18821; 
New  York,  219609 ;  Boston,  75065  ;  Philadelphia,  58632  ; 
Various,  57976.  How  many  bales  were  received  during 
the  year?  Ans,  4127892, 


Examples  in  Addition.  29 

•  84.     From  Aug.  31st  1875  to  Sept.  1st  1876,  the  pro- 
duction of  Sugar  in  Louisiana  was  as  follows  : 

Parish  of  Livingston,  4  hogsheads;  St.  Tammany,  16; 
East  Feliciana,  37  ;  Lafayette,  187  ;  West  Feliciana,  339  ; 
Vermillion,  609  ;  Avoyelles,  1582  ;  St.  Landry,  1768  ;  St. 
Martin,  1884;  Orleans,  1041;  St.  Bernard,  2097;  East 
Baton  Rouge,  2544  ;  Rapides,  2453  ;  Pointe  Coupee,  2762  ; 
Iberia,  3632;  Jefferson,  3671  ;  West  Baton  Rouge,  4155  ; 
St.  Charles,  5808;  St.  John,  8335;  Plaquemines,  9068; 
Iberville,  9814;  Lafourche,  11302;  Terrebonne,  10888; 
St.  James,  13437;  Ascension,  H267  ;  St.  Mary,  14318 ; 
Assumption,  14712.  How  many  hogsheads  were  produced 
during  the  year  ?  Ans.  140730. 

85.  From  July  1st  1875  to  July  1st  1876,  the  monthly 
receipts  of  coffee  in  New  Orleans,  were  as  follows : 

July,  9635  bags  ;  August,  25987  ;  September,  24851  ; 
October,  8832  ;  November,  34452  ;  December,  4800 ;  Jan- 
uary, 32219;  February,  16042,  March,  4000;  April, 
10512  ;  May,  9000  ;  June,  15  120.  How  many  bags  were 
received  during  the  year?  Ans.  195450. 

86.  From  New  Orleans  to  Carrolton  is  7  miles;  hence 
to  Donaldsonville,  71  ;  hence  to  Pln^m-mines.  32  ;  hence  to 
Baton  Rouge,  20;  hence  to   Port   Hudson,  23;  hence  to 
Bayou  Sara,  12  ;  hence  to  mouth  Red  River,  40  ;  hence  to 
Natchez,  72  ;  hence  to  Rodney,  45  ;'  hence  to  Grand  Gulf, 
18  ;  hence  to  Vicksburg,  61  ;  hence  to  the  Louisiana  Line, 
97 ;  hence  to  Helena,  230  ;  hence  to  Columbus,  329  ;  hence  to 
Cairo,  20  ;  hence  to  (/ape  Ginwdeau,  50  ;  hence  to  St.  Louis, 
151.      How  many  miles  to  St.  Louis  by  river? 

Ans.  127S  miles. 

87.  From    New  Orleans  to  tin;  mouth  of  Red  River  is 
210  miles  ;  hence  to  Black  River,  40  ;  hence  to  Alexandria, 
110;  hence  to  Grand  Ecore,  120;  hence  to  Grand  Bayou, 
95  ;  hence  to  New  Hope,  60  ;  hence  to  Waterloo,  30  ;  hence 
to  Shreveport,  35.     How  many  miles  to  Shreveport  by  the 
river?  Ans.  700  miles. 

88.  From  New  Orleans  to  Algiers  Depot  is  1  mile . 
hence  to  Gretna,  3 ;  hence  to  Jefferson,  9 ;  hence  to  St'. 

v* 


30          Arithmetical  Exercises  and  Examples. 

Charles,  0;  hence  to  Bouttc,  6;  hence  to  Huyou  des 
Alemedes,  8;  hence  to  Kaeeland,  S  ;  henee  to  Kwin 
hence  to  Lafourche,  0  ;  henue  to  Tenvbonne,  -> :  hemv  t«» 
Chucahoula,  ft  ;  hence  to  Tigerville.  5  ;  IUMICC  to  L'0;i  8  -. 
4;  hence  to  Bayou  Beuf,  3;  hence  to  Ramos,  3 :  benee  to 
Morgan  City,  4;  hence  to  Galveston,  240.  How  many 
miles  to  Ualveston  ?  A  us.  321  miles. 

89.  24  peaches  were  eaten,  5  being  spoiled,  were  thrown 
away,  and  32  remained  in  the  basket.     How  many  were 
there  at  first  ?  A  us.  (51. 

90.  A  man  was  2H  years  of  age  when  lie  was  married. 
TIow  old  will  he  be  when  he  has  been  married  1  I  years  ? 

AIIS.     1<>  years. 

!H.  A  young  man  graduated  from  college  when  he  was 
22  years  of  age.  lie  married  l>  years  afterward.-.  2  year- 
after  that  he  was  presented  with  a  son.  What  will  he  his 
age  when  the  son  is  21  years  old?  A  us.  ,">!  year<. 

92.  A  lady  paid  §(f.50  for  a  dress,   $8  for  a  shawl,  $1 
for  a  bonnet  and  83.75  for  a  pair  of  shoes.     What  was  tho 
total  cost?  Ans.  822.2."). 

93.  A  boy  sold  his  pony  for  $45,  and  lost  §15  by  the 
sale.     What  did  the  pony  cost  him?  Ans.  $60. 

94.  A  merchant  paid  for  a  lot  of  goods  $580,  he  sold 
them  and  gained  $190.     How  much  did  he  receive  for 
them?  Ans.  8770. 

95.  Henry  is  16  years  old,  James  is  3  years  older,  and 
William  is  2  years  older  than  James.     How  old  are  James 
and  William?  Ans.  James  1(J,  William  21. 

96.  The  internal  framework  of  the  human  body  con- 
sists of  bones,  which  united  by  strong  ligaments  constitute  the 
skeleton.     In  the  skull  are  8  bones;  in  the  face  14  ;  in  each 
ear  3  ;  in  the  tongue  1  ;  in  the  trunk  and  spinal   column 
and  pelvis  55  ;    in  each    shoulder  2  ;  in  each  arm   3  ;  in 
each  wrist  8 ;  in  the  palm  of  each  hand  5  ;  in  each  thumb 

2  ;  in  each  finger  3  ;  in  each  leg  4 ;   in  each  ank'e  7  ;  in 
each  foot  5  ;  in  each  great  toe  2  ;  in  each  of  the  other  toes 

3  ;  and  there  are_  32  teeth.     How  inaiiy  bones  in  the  whole 
body?  Ans.  240, 


Examples  in  Addition.  31 

97.  How  many  pupils  in  a  school  in  which  there  are  6 
grades,  the  first  containing  63,  the  second  58,  the  third  27, 
the  fourth  41),  the  fifth  35  and  the  sixth  24?  Ans.  256. 

98.  Bothick   has   $420;  Conrad  has  $130  more  than 
Bothick,  and  Prophet  has  as  much  as  Bothick  and  Conrad 
together.      What  sum  have  all  three  ?  Ans.  $1940. 

99.  Keen,   Soule    and  Abbott  form    a    copartnership, 
Keen  invests  83400,  Soule  $4000,  and  Abbott  $500  more 
than  both  Keen  and  Soule.     What  is  the  capital  of  the 
linn?  Ans.  $15,300. 

100.  A  father  gave  his  son  seven  thousand  eight  hun- 
dred dollars  ;  his  daughter  nineteen  hundred  and  fifty  dol- 
lars ;  and  his  wife  three  thousand  five  hundred  more  than 
he  gave  to  both,  the  son  and  daughter.      What  sum  did  he 
give  away  ?-  Ans.  $23,000. 


SUBTRACTION,  (Decreasing.) 


39.  Subtraction  is  the  process  or  operation  of  finding 
the  difference  between  two  numbers  of  the  same  kind. 

40.  The  result  obtained  by  subtraction   is  called   the 

Difference  or.Remainder. 

41.  The  greater  number  is  called  the  Minuend,  which 
means  a  number  to  be  decreased. 

4'J.  The  lesser  number  is  called  the  Subtrahend,  winch 
moans  the  number  to  be  subtracted. 

4:>.  The  sijjn  of  subtraction  is  a  horizontal  line,  — . 
It  is  read  minus  and  means  less. 

When  this  sign  is  placed  between  two  numbers  it  nu,in> 
that  the  number  nfte.r  it,  is  to  be  subtracted  trom  the  nuin- 
I'cr  Injure  it.  Thus  8  —  3  is  read  8  minus  3. 

For  Subtraction  Tables  and  contracted  method^  of  sub^ 
fraction  see  Soule's  Contractions  in  Numbers. 


32          Arithmetical  Examples  and  Exercises. 

\  I.  The  sign,  (  ),  parenthesis,  or ,  vinculum,  indi- 
cate that  the  numbers  included  within  the  parenthesis,  or 
below  the  vinculum,  are  to  be  considered  as  one,  or  together. 
Thus  (9+3) — 5—7,  or  with  the  vinculum  thus  9+3 — 5 

—7. 

45.  ORAL  EXERCISES. 

1.  Commence  at  50  and  orally  count  to  0  by  continually 
subtracting  1,  thus :  49,  48,  47,  46,  45,  etc. 

2.  Commence  at  50  and  orally  count  to  0  by  continually 
subtracting  2,  thus :  48,  46,  44,  42,  etc. 

3.  Commence  at  50  and  orally  count  to  0  by  successively 
subtracting  3,  thus  47,  44,  41,  38,  etc. 

4.  In  like  manner  commence  at  50  and  subtract  respect- 
ively 4,  5,  6,  7,  8,  9  and  10  until  you  produce  0  or  a  num- 
ber less  than  the  subtracted  number,  thus  46,  41,  35,  28, 
etc. 

5.  Commence  at  50  and  subtract  alternately  2  and  5 
until  you  produce  1,  thus,  48,  43,  41,  36,  etc. 

6.  Commence  at  50  and  subtract  alternately  8  and  3 
until  you  produce  6,  thus  42,  39,  31,  etc. 

46.  To  subtract  one  number  from  another   irhrti  any 
figure  of  the  subtrahend  is  less  than  the  corretpondiitg  figure 

of  the  minuend. 

1.     From  897  subtract  (>41. 

OPERATION. 

897  641  Explanation. — First  set  the  numbers 

p}^|    or    gQ-7  with  the  less  under  or  over  the   greater, 

so  that  units   of  the   same  order    \\ill 

~  ,  stand  in  the  same  column.     Then  com- 

2o6  Lob  mence  with  the  units  figure  and  subtract 

each  order  separately  ;  thus,  1  from  7  leaves  6  ;  4  from  9  leaves 

5  ;  6  from  8  leaves  2.     By  this  work  we  obtain  the  difference  or 

remainder,  256. 

Subtract  the  following : 

Gi)        (:*)  W  (5)  (6) 

843     384  978  425  9876 

521     762  655  679  3456 


Subtraction — Decreasing.  33 

47.  1o  subtract  one  number  from  another  when  any 
figure  of  the  subtrahend  is  greater  than  the  corresponding 
figure  of  the  minuend. 

1.     From  4173  subtract  2345. 

FIRST    OPERATION. 


Minuend         4173  Subtrahend     2346 

Subtrahend     2346  Minuend         4173 


Difference       1827  Difference       1827 

Explanation. — Having  written  the  numbers  as  in  the  preceding 
problem,  with  the  lesser  number  either  above  or  below  the 
greater,  we  observe  that  6  units  cannot  be  taken  from  3  units  ; 
we  therefore  mentally  add  10  to  the  3  units  making  13  units, 
and  then  say  6  from  13  leaves  7;  then  as  we  added  10  to  the 
minnend  we  now  mentally  add  its  equivalent,  1  ten,  to  the  tens 
figure  of  the  subtrahend,  and  say  5  from  7  leaves  2  ;  we  next 
observe  that  3  hundreds  cannot  be  taken  from  1  hundred,  we 
therefore  mentally  add  10  hundreds  to  the  1  hundred  making  11 
hundreds,  and  then  say  !'•  from  11  leaves  8  ;  then  having  added 
10  hundreds  to  the  hundreds  figure  of  the  minuend  we  now 
mentally  add  I  thousand,  the  equivalent  of  the  10  hundreds,  to 
the  thousands  figure  of  the  subtrahend  and  say  3  from  4  leaves 
1.  This  completes  the  operation  and  gives  1827  as  the  difference 
of  the  two  numbers. 

The  addition  of  10  to  the  units  and  10  hundreds  to  the  hun- 
dreds of  the  minuend,  and  its  equivalent  1  ten  and  1  thousand 
to  the  tens  and  thousands  columns  of  the  subtrahend,  is  done 
upon  the  principle  that  the  difference  between  two  numbers  is 
the  same  as  the  difference  between  the  same  two  numbers 
f^unllti  incrcasi'd. 

In  all  problems  of  subtraction  the  operation  of  adding  10  to 
the  minuend  and  its  equivalent,  1,  of  the  next  higher  order  to 
the  subtrahend  is  repeated  as  often  as  the  subtrahend  figure  is 
greater  than  its  corresponding  minuend  figure. 

To  Prove  subtraction  add  the  difference  or  remainder  to 
the  subtrahend  and  if  the  sum  is  equal  to  the  -minuend  the 
work  may  be  considered  correct, 


Arithmetical  Exercises  and  Examples. 


SECOND  OPERATION. 
•11  To 

-117:5 


1Q97 

1<s-' 


Explanation.  We  will  here  perform 
the  operation  by  addition  which  is 
a  simpler  and  better  method  than 
the  preceding,  and  consists  simply 
in  a(J(linK  to  tlie  subtrahend  such  a 
number  as  will  make  it  equal  to  the 


minuend.  Thus  commencing  with  the  unit  figure  of  the  sub- 
trahend or  smaller  number,  we  say  6  and  ?  make  13;  and  set 
the  7  in  the  units  place  of  the  difference;  then  carrying  1  we 
say  5  and  2  make  7,  and  set  the  2  in  the  tens  column  of  the 
difference  ;  then  we  say  3  and  8  make  11,  and  write  the  8  in  the 
third  column  or  hundreds  place  of  the  difference;  then  carry- 
ing 1  we  say  3  and  1  make  4,  and  write  the  1  in  the  fourth  place 
of  the  difference.  This  completes  the  operation. 

2.      From  7:»215  subtract  122*. 

FIRST  OPERATION. 

fj\^>  j  -  Explanation*     Here  we  say  8  from  15 

, .".,'  leaves  7  ;  3  from  4  leaves  1  ;   2  from  2 

leaves  0;   1  from  3  leaves  2  ;  0  from  7 
leaves  7. 

72017 

SECOND  OPERATION. 

73245 

122S 


72017 


Explanation.  Here  we  say  8  and  7 
make  15;  3  and  1  make  4;  2  and  0 
make  2 ;  1  and  2  make  3 ;  0  and  7 
make  7. 


3.     From  56802  subtract  50531. 


FIRST  OPERATION. 

56802 
50531 


Explanation. — Here  we  say  1  from  2 
1  ;  3  from  10,  7  ;  G  from  8,  2  ;  0  from 
6,  6  ;  5  from  5,  0,  which  being  the  last 
figure  on  the  left  has  no  value,  and 
hence  is  not  set. 

Explanation. — Here  we  say  1  and  1 
—2  ;  3  and  7—10  ;  0  and  2  =8  ;  0  and 
6=6  ;  5  and  0=5.  The  naught  is  not 
set  for  the  reason  given  in  the  first 
solution. 

EXAMPLES. 

Write  the  following  groups  of  numbers  as  they  are  here 


6271 

SECOND  OPERATION. 

56802 
50531 


6271 


Subtraction.  31> 

written  and  subtract  the  lesser  from  the  greater  of  each 
group : 

467           1807           3842            607  3001          6879 

342           4251           1291          8013  1009          9640 


Subtract  the  following  numbers : 

7.  From  5307  take  309.  Ans.  4998. 

8.  From  1090  take  1009.  Ans.  81. 

9.  From  7608  take  3705.  Ans.  3903. 

10.  From  184240  take  39460.  Ans.  144780. 

11.  From  41074089  take  1875429. 

Ans.  39198G60. 

12.  From  9876543210  take  123456781)0. 

Ans.  8641975320. 

48.      To  Subtract  Dollars  and  Cents. 

1.     What  is  the  difference  between  $483  and  $51.65. 

Ans.  $431.35. 

OPERATION.  Explanation. — In  all  problems  of  this 

$483.00  kind  we  first  set  the  numbers  in  the 

.-  i\.r  same  manner  as  when  adding  dollars 

and  cents,  with  dollars  under  dollars 

and  cents  under  cents,  so  that  units  of 

$431.35  the  same  order  will  stand  in  the  same 

column  and  the  points  in  a  vertical  line. 

When  there  are  no  cents  in  the   minuend,  we  fill  the 
place  of  cents  with  naughts. 

The  operation  of  subtraction  is  performed  with  dollars 
and  cents,  the  same  as  with  other  numbers. 

What  is  the  difference  between  the  numbers  in  each  of 
the  following  groups  ? 

$16.25        $8~00        $.75        $4l!o4        $10.50        $L93 
9.38          3.75          .59  6.61  4.78  .47 


$6.87       $4.25 


*3ft          Arithmetical  Exercises  and  Example*. 

$681.85        $127.05        $24S.oo        silUl         $8527.09 
90.38  lo:>.r>n  181.15  !>.s<> 


12    •  1:;  14  IT, 

875.00  1971.50  lti-10.10  5184.62 

43.50  >.oo  1270.UO  52!). 7S 


10.     Paid  for  rice  85500,  and  fur  sugar  80875.40.      1 1  «>w 
much  more  was  paid  for  sugar  than  rice  ? 

Ans.     §1375.10 

17.      Bought  a  lot  of  flour  for  $2225,  and  sold  the  same 
for  8'2SOO,  what  was  the  gain  ?  ADS.      $575. 

IS.      It  is  700  niilos  to  Shivveport  and  320  to  (Jalveston. 
How  much  farther  is  it  to  Shivvepurt    than  to  (Jalveston? 

Ans.     3SO  miles. 

10.     The  ant  has  fifty  eyes,  the  dra.-on  fly  12000  ;   how 
many  more  has  the  dragon  fly  than  the  ant  ? 

Ans.     11 950  eyes. 

20.  The  total  coinage  of  gold  and  silver  at  the  different 
mints  of  the  U.  8.  during  the  fiscal  year  ending  June  30th 
1875,  was  843,854,708.    Of  thls-aniount  $33,553,965  was 
g<  Id,  what  was  the  amount  of  silver  coined. 

Ans.     $10.300,743. 

21.  A  student  had  40  problems  to  work,   he  worked 
17,  how  many  has  he  yet  to  work  ?  Ans.     23. 

22.  Man   has  26  bones  in  each   foot,   and   1!>  in  each 
hand,  how  many  more  has  he  in  the  foot  than  in  the  hand  ? 

Ans.     7. 

23.  Sound  travels  through   the  air  at  the  rate  of  1118 
feet  per  second,  and  a  bullet  fired  from  a  rifle  travels  1750 
feet  per  second  ;  how  much  faster  does  the  ball  travel  than 
sound  ?  Ans.     632  feet  per  second. 

24.  Physiologists  have  determined  with  the  aid  of  the 
microscope,  that  the  lungs  of  man  contain  not  less  than 
600,000,000  air  cells  ;  they  have  also  determined  that  a  sin- 
gle drop  of  human  blood  contains  more  than  4jOOOjOOO,000 


Subtraction.  37 

of  corpuscles ;  how  many  more  corpuscles  in  one  drop  of 
blood  than  air  cells  in  the  lungs  ?      Ans.  3,400,000,000. 

25.  Geologists  have  demonstrated  that  the  formation  of 
the  stalactites  and  stalagmites  in  the  Mammoth  Cave  of 
Kentucky,  required  not  less  than  75000  years  of  time  ;  and 
that  the  wearing  away  of  the  rock  of  Niagara  Falls  by 
friction,  from  Queenstown  where  they  first  were  after  the 
glacial  epoch,  to  their  present  location,  7  miles  above, 
required  at  least  40,000  years ;  how  much  longer  did  it 
require  to  form  the  stalactites  and  stalagmites,  than  for  the 
Falls  of  Niagara  to  recede  to  their  present  location? 

Ans.     35000  years. 

How  many  years  from  the  date  of  each  of  the  following 
events  to  the  present  year  ? 

20.     Quills  first  used  for  writing   IJ30  A.  I>. 

27.  Figures  used  by  the  Arabs,  borrowed  from  the 
Indians,  813  A.  D. 

2H.      High  towers  first  erected  on  churches,  1000  A.  D. 

2!).      Glass  Windows  first  used  in  Kn-land,    11  SO  A.  D. 

30.  Chimneys  built  in  Kngland,  1230  A.  D. 

31.  Spectacles  invented  by  Spinu.  1299  A.  D. 
32^_\Voolen  cloths  first  made  in  England,  1331  A.  D. 

33.  Muskets  used  in  .Knglaml.  1421,  A.  D. 

34.  Printing  invented,  14  K)  A.  D. 

35.  Almanacs  first  published  in  Hilda,  1400  A.  D. 
30.  Tobacco  discovered  in  St.  Domingo,  1490  A.  D. 
37.  Spinning- wheel  invented  at  Brunswick,  1530  A.  D. 
33.  Needles  first  made  in  Kniiland   by  an  Kast  Indian, 

1545  A.  D. 

39.  Decimal  Arithmetic  invented  at  Bruges,  1602  A.  D. 

40.  Circulation   of  the  blood   discovered   by  Harvey, 
1619  A.  D. 

41.  Newspapers  first  published,  1030  A.  D. 

42.  Coffee  brought  to  England,  1041  A.  D. 

43.  Steam  engines  invented  by  the  Marquis  of  Wor- 
cester, 1049  A.  D. 

44.  Cotton  first  planted  in  the  United  States.  1709  A. 
D. 


38'        Arithmetical  Exercises  and  Examples. 

45.      Cotton  first  spun  in  America.  1787  A.  D. 

4G.  Steam  first  used  to  propel  boats  by  Fulton,  in  Amer- 
ica, 1807  A.  D. 

47.  First  Locomotive  was  made  at  Liverpool,  1820  A. 
D. 

H.  Electro-Magnetic  Telegraphy  invented  by  Morse, 
of  America,  IS.'!:*  A.  D. 

4(J.      America  was  discovered  in  141)2  A.  1). 

50.  The  electric  telegraphy  was  first  used  in  the  Tinted 
States  in  1844  A.  D. 

51.  (Jem-rul  (Jeorge.  Washington  was  burn  in  1732  and 
died  in    171W;   (Jeneral   11.  H.  Lee  was  born  in  1807  and 
died  in  1870.      How  much  older  was  Ueiieral  Washington 
thaa  General  Lee,  when  he  died?  Ans.  4  years. 

52.  What  is  the  difference  between  23222  and  11  thou- 
sand 11  hundred  and  1 1  ?  Ans.  11111. 

53.  What  is  the  difference  between  0  dozen  dozen  and 
half  a  dozen  dozen  ?  Ans.  792. 

54.  What  number  must  be  ndded  to  G8741  to  make  a 
million?  Ans.  931251). 

55.  Philadelphia  has  153151  buildings;  New  Orleans 
35600.     How    many   more   has   Philadelphia    than    New 
Orleans?  Ans.  117551. 

5G.  James,  who  is  23,  is  7  years  older  than  Henry; 
how  old  is  Henry?  Ans.  lb'  years. 

57.  William  has  8500  which  is  $150  more  than  I,  and 
I  have  $75  more  than   Lewis,  how  much  has  Lewis,  and 
how  much  have  I  ?  Ans.  Lewis  has  $275. 

I  have  $350. 

58.  There  are*  two  parties  who  owe  me  $8000,  one  of 
them  owes  $4250.     The  other  wishes  to  pay  me  $1700  on 
account ;  how  much  will  he  then  owe  ? 

Ans.  $2050. 

59.  A  speculator  bought  a  lot  of  apples  for  $215.,  and 
sold  them  for  such  a  price,  that  if  he  had  got  $22.50  more 
he  would  have  gained  as  much  as  they  cost  him.     How 
much  did  he  sell  them  for  ?  Ans.  $407.50. 

60.  From  New  Orleans  to  Vicksburg  is  401  miles,  and 


Subtraction.  39 

to  Natchez  277  miles  ;  how  far  is  it  from  Natchez  to  Vicks- 
burg?  Ans.   124  miles 

61.  What  is  the  difference  between  one  million,  seven- 
teen thousand  and  seven,  and  one  thousand  sixteen  hundred 
and  sixteen  ?  Ans.  1,014,391. 

62.  The  sum  of  two  numbers  is  1463,  one  of  the  num- 
bers is  628,  what  is  the  other  ?  Ans.  835. 

63.  The  velocity  of  our  earth   on  its  yearly  voyage 
through  space,  around  the  sun,  is  99733  feet  per  second  ; 
the  velocity  of  a  12  pound  cannon  ball  fired  from  a  gun 
with  an  average  charge  of  powder  is  1734  feet  per  second, 
how  many  feet  farther  does  the  earth  travel,  in  each  second, 
than  a  cannon  ball  ?  Ans.  97999  feet, 

or  18  miles  and  2959  feet. 

64.  What  number  is  that  to  which  if  17821  be  added 
the  sum  will  be  37907  ?  Ans.  20086. 

65.  At    an    election   the    defeated    candidate    received 
23742  votes;  had  he  received  5112  votes  more  he  would 
have  been  elected  by  1000  majority;  how  many  votes  did 
the  elected  candidate  receive  ?  Ans.   278.")  I. 

66.  A  father  divided  his  plantation  consisting  of  4500 
acres    between    his    five   sons    Albert,    Edward,    William. 
Frank   and   Robert.     To   Albert   he   ^ave   SOO   acres;  to 
Edward  he  gave  150  acres  more  than   he  gave  Albert;  to 
William  he  gave  100  acres  less  than  he  gave  Edward;  to 
Frank  he  gave  as  much  as  he  gave    I'M  ward,  and  the  re- 
mainder he  gave  to  Robert.     How  many  acres  did  Robert 
receive?  Ans.  950  acres. 

I 


40          Arithmetical  Exercises  and  Examples. 


MULTIPLIC  ATIO>-( 

-W.  Multiplication  is  the  processor  operation  of  in- 
creasing one  of  two  numbers  as  many  times  as  there  are 
units  in  the  other.  Or,  differently  explained,  it  is  a  short 
method  of  performing  addition. 

The  number  to  be  multiplied  is  called   the  multiplicand. 

The  number  which  shows  how  many  times  the  multipli- 
cand is  to  be  increased  or  repeated  is  called  the  innlfi/t/irr. 

The  result  obtained  by  the  operation  of  mttltiplyiog  is 
called  the  Product. 

The  multiplicand  and  multiplier  are  railed  furffirs.  The 
meaning  of  the  word  factor  is  •//< ///,•/_•/•  or  pm<lnc<  r. 

50.  The  Sigll  Of  Multiplication  is  an  oblique  cross, 
X-   It  is  read  multiplied  !>>/  or  fiims.    Thus  8  ><  •>,  is  read 
8  multiplied  by  3,  or  3  times  8. 

51.  Principles  of  Multiplication.    In  all  cases  of 
multiplication  the  multiplier  must -be  regarded  as  an  abstract 
number.     Two  denominate  numbers  cannot  be  multiplied 
together  as  denominate  numbers. 

In  all  multiplication  operations  the  product  is  the  x<nn<'  in 
name  or  kind  as  the  multiplicand. 

For  extended  work  on  contracted  methods  of  multiplying 
see  Soule's  Contractions  in  Numbers. 


Multiplication  Table. 


41 


112181  4|     5|     6|     7|     8|     9|  10 


2|  4 1  6|  8|  10]  12|  14j  T6|  18|  20 


3|  6}  9|12|  15|  18|  21|  24 1  27|  30 


4]_8J12|16|_20J_24J  2s  :',2|  3G|  40 

5J10J15J20    2--» |  30 1  36|  40|  45|_50 

C.  12J18J24J   30|  3G|  42j  48|  54;  (id 

7  II  21J28    35]  42|  49]  56|  03 1  70 

sin|24|32|    Hi  48  .'.(•. ;''64|~72|80 

"9|18|2"7|3l5|  40|  54j  fir!;  72;  M|~90 

10[20|30J40|  H(!|  G0|  7d|  8d|  9o;i«nn 

II  22|33  II    55j  66  77  8_8    99J110 

1224|:-{r.|4S|  Gdj  72|  84|  96|108|120 


13|2j6|39j52|  G5| 

14|28|42|56|  7(l| 

15)30  |.Vi.i«ij  7.">j 

16J32  \>-  M  80   '.MM  1  1-  12s  144J160 

17J3J  5J  6J  85 


78|  91J104J1I7130 

84|  !)8J112]l2G|140 
'.MI  I(i:.jl20|13."i 


r,-{|l70 

18J36J5472  '••  .....  8(126J144J162|180 
L9|38J57|76  96  114J133J152  I71|190 
20]40|6Tl|80  KHI  l^n  I  1(1,1(11)  ISII^IKI 


Erplanfilion  — WP  rrcomnipnd  tliif*  taMp  as  being  far  superior  to  the  one  pre- 
Bftitod  in  the  School  and  College  Text  Nooks  of  the  country,  and  urge  all  who 
a«pire  to  proficiency  in  computing  numl»ci>  to  learn  it.  In  learning  tint  table, 
or  in  the  use  of  it.  we  caution  the  calculator  apainet  the  use  of  all  intermediate 
•words,  whether  he  speak.*  or  thinks  them;  thus,  instead  of  saying  or  thinking, 
(J  times  3  are  27;  17  thm*  r,  are  lu^,  AT.,  say  or  think,  9,  3,  27;  17,  6,  102,  <tc. 

Tn  reading  we  do  not  stop  to  spell  orally  or  mentally  the  words  that  compose 
the  sentences;  from  the  combination  of  the  letters  we  see  what  the  words  are 
without  looking  specially  at  each  individual  letter,  and  to  read  or  operate  with 
rapidity  in  the  combination  of  numbers,  we  muHt  omit  all  superfluous  talk  or 
Ih  ought.. 

P* 


42          Arithmetical  Exercises  and  Examples. 
52.    ORAL   EXERCISES. 

1.  At  8  cents  a  pound  what  will  9  pounds  of  rice  cost  ? 

Solution. — In  all  problems  of  this  kind  we  reason  thus  :  Since 
1  pound  costs  8/,  9  pounds  will  cost  9  times  us  much,  which  is 

w, 

2.  What  will  7  yards  cost  at  12/  per  yard  ? 

3.  What  will  4  books  cost  at  20/  each  ? 

4.  At  13/  per  dozen  what  will  6  dozen  cost  ? 

5.  If  1  box  (•'  -i  |3  what  will  23  boxes  cost? 

6.  Flour  is  worth  $7  per  barrel,  what  are  25  barrels 
worth  ? 

7.  Bought  14  pounds  of  sugar  at  8/  per  pound,  what 
did  it  cost  ? 

8.  At  *i>  per  curd  what  will  34  cords  of  wood  oos(  ? 

9.  Paid  $4  per  barrel  for  potatoes  and  bought  47  bar- 
rels, what  did  they  cost? 

10.  If  you  receive  §2   per  day  for  labor  and  work  17 
days  how  much  money  will  you  receive  ? 

Solution. — Since  1  day's  labor  is  worth  $2,  17  days'  labor  is 
worth  17  times  as  much,  which  is  $3i.  Or  thus,  since  I  receive 
$2  for  1  day's  work,  for  17  days'  work  I  will  receive  17  limes  as 
much,  which  is  $34. 

11.  Multiply  from  0  times  8  to  15  times  8  and  reverse. 

12.  Multiply  from  0  times  9  to  16  times  9  and  reverse. 

13.  Multiply  from  0  times  11  to  16  times  11  and  reverse. 

14.  12  inches  make  a  foot,  how  many  inches  in  16  feet  ? 

15.  4  quarts  mako  a  gallon.     How  many  quarts  in  a 
barrel  that  holds  42  gallons  ? 

16.  What  will  6  dozen  shirts  cost  @  $18  per  dozen  ? 

17.  If  you  buy  15  boxes  of  peaches  (a}  $2  per  box, 
what  will  they  cost  ? 

18.  Multiply  from   0  times   12   to   19   times  12   and 
reverse. 

19.  How  many  are  9  times  12  plus  8? 

20.  How  many  are  12  times  7  minus  6  ? 

21.  If  you  buy  7  pencils  at  5  cents  each  and  hand  to 


Multiplication.  43 

the  seller  50^,  how  much  change  ought  you  to  receive  ? 

22.  A  merchant  bought  23  barrels  of  apples  at  $4  per 
barrel  and  paid  $65  on  account.  How  much  does  he  still 
owe  ? 

53.  To  multi.ply  whwi  the  multiplier  consists  of  only  <»«: 
figure. 

1.     What  is  the  product  of  947  multiplied  by  6  ? 

OPERATION.  Krplanation. — In  all  problems  of  this 

=  r:^  kind  we  place  the  multiplier  under  the 

E.7  7  units'  figure  of  the  multiplicand   and 

i.  then  commencing  with  the  units  figure 

we  say,  6  times  7  are  42,  which  is  4 

Multiplicand   947  /<//*  and  2  unite;  the  2   unit*  we  place 

Multiplier  (')  i"  the  units  place  of  the  product  and 

retain  in  the  mind  the  4  tens  to  add 

Prnrinff       ^<N«>        to  the  column  of  tens;  we  next  sa.v 

6  times  4  are  24  plus  the  4  tens  re- 
tained in  the  mind,  are  28,  which  is  2  hundreds  and  8  *<•«*,  the  8 
/<;/.v-we  write  in  the  tens  column  ot  the  product,  and  retain  in 
the  mind  the  2  hundreds  to  add  to  the  column  of  hundreds. 
We  then  say  (>  times  t>  are  54.  plus  2  hundreds  are  ,"»G.  which  is 
5  thousand  and  •'>  lnnnln<lt,  wliich  we  write  respectively  in  the 
thousands  and  hundreds  columns  of  the  product.  This  com- 
pletes the  operation  and  gives  a  product  of  .> 

In  practice,  instead  of  saying  U  times  7  are  42,  G  times  4  are 
24,  etc.,  we  should  only  name  the  result  of  the  combination, 
thus  42,  24,  etc.  In  handling  fif/ur?x  NV  slum/if  /////w//,s  j>ron<nin<-r 
t':<  results  of  the  combinations  trithout  )unni><</  the  flour's  tlxtt  Jitaht: 
the  rryii1t,jm<t  <ts  ice  pronounce  words  without  xpclliny  or  naminj  the 
Utters  that  make  the  word*. 

">•!.    ToProvo  the  operations  of  multiplication,  repeat  the 

work  or  multiply  the  multiplier  by  the  multiplicand.  If 
the  result  is  the  same  as  the  first,  the  work  is  probably 
correct. 

EXAMPLES. 

Perform  the  following  multiplications: 
•j  -i 

Multiplicand  5t:j  '.)>:;  27(i!)  7<iS!)5 

3Lultiplicr  785  9 


Product    :;S()1     7864     13845     692055 


44          Arithmetical  Examples  and  Exercises. 


s7<;  t 

5 

2987 

8 

;<; 

7 

<J 

10 

46532 
14 

11 
58(17  I 
15 

ia 
9861 

17 

18 

si  i:,:i 
19 

14.  What  will  4  pianos  cost  at  $425  each  ? 

Ans.  81700. 

15.  At  $65  each  what  will  0  wagons  cost  ? 

Ans.  |585, 

16.  What  will  7  lots  of  ground  cost  at  $187")  each  ? 

Ans.  $13125. 

17.  At  $6  per  barrel  what  will  be  the  cost  of  245  bar- 
rels of  flour? 

OPERATION.  /•.'.r/'litnution. — The  analysis  for  this 

Multiplier         'Mf>  problem  is,  since  1  barrel  of  Hour 

',,  i    "  cost  $6,    245   barrels   will  cost   245 

MulUplicand         I,  times  as  much.     The  $6  is  the  real 

multiplicand,   but  in   the  operation 
#1470    Ans.   we  used  it  as  the  multiplier.     This 
wt>  do  for  convenience   in  performing  the  operation,  in   all  pro- 
blems where  the  multiplicand  is  less  than  the  multiplier.      The 
result  is  the  same  whichever  factor  we  use  as  a  multiplier. 

18.  What  will  42  dozen  hats  cost  at  $9  per  dnzen  ? 

Ans. 

TJ.     At  $7  a  piece  what  will  48  chairs  cost  ? 

A  us. 


Multiplication.  45 

55.      To  multiply  when  the  multiplier  consists  of  more 
than  one  figure. 

1.     What  is  the  product  of  397  multiplied  by  653  ? 


. 

•S*|  =  *  s 

ss  §  .=  =  ?  '3 

Multiplicand   '   397 
Multiplier  653 

1st  Partial  product  by  3  units   11  $\=3  times  the  multiplicand. 
2d  Partial  product  by  5  tens  1985   =50  times  " 

3d         "  "        6  h'ds-   2382      =600         "  " 


Total  product          259,241=653         " 

Ksplanation. — In  all  problems  of  this  kind  we  first  write  the 
multiplier  under  the  multiplicand  so  that  units  of  the  same 
order  will  stand  in  the  same  column,  and  then  multiply  by  one 
figure  at  a  time.  We  first  multiply  by  the  units  figure,  then  the 
/<•;/.*,  hurn/rrt/A  and  so  on  in  regular  order  through  the  multiplier 
and  add  the  several  partial  products  together  and  thus  obtain 
the  required  product. 

In  this  problem  we  first  multiply  by  .?,  the  units  figure,  in  the 
same  manner  as  explained  in  the  first  problem  where  there  was 
but  one  figure  in  the  multiplier,  and  obtain  1191  as  the  first 
partial  product.  This  we  write  below  the  multiplier  so  that 
units  of  the  same  order  will  stand  in  the  same  column. 

Next  we  multiply  by  the  5  tens  ;  we  say  5  times  7  are  35,  which 
is  3  hundreds  and  5  tens  ;  we  write  the  5  tens  in  the  tens  column 
directly  below  the  multiplying  figure  and  reierve  in  the  mind 
the  3  hundreds  to  add  to  the  hundreds  column.  We  then  say 
5  times  9  are  45  -f-  3  hundreds  which  were  reserved  are  48  hun- 
dreds which  is  4  thousands  and  8  hundreds;  we  write  the  8 
hundreds  in  the  column  of  hundreds  and  reserve  the  4  thous- 
ands to  add  to  the  thousands  column.  We  then  say  5  times  3 
are  15  plus  4  thousands,  reserved,  are  19  thousands,  which  is  1 
ten  thousand  and  9  thousands,  which  we  write  in  tlmir  respect- 
ive columns. 

We  then  in  like  manner  multiply  by  the  6  hundreds  in  the 
multiplier,  being  careful  to  write  the  first  figure  obtained  (2) 
in  the  hundreds  column,  directly  under  the  G  of  the  multiplier, 
and  the  other  figures  in  their  respective  columns,  thousands,  ten 


K  >          Arithmetical  Exercises  and  Examples. 


th<mxan<h  and  hnndr«1  th»u*andx.  We  then  add  the  partial  pro- 
ducts together  and  obtain  259241  as  the  whole  product  of  397 
multiplied  by  653. 

In  practice  remember  to  name  or  think  only  the  results  of  the 
numerical  combinations  when  adding  or  multiplying. 

EXAMPLES. 

2.     Multiply  3426  by  457. 

OPERATION. 

11. 


Multiplicand 
Multiplier 


3426 
457 


1.  Partial  prod,  by  7  units  23982=  7  times  the  multiplicand 

2.  Partial  prod,  by  5  Uns  17130     =3  times   the  multiplicand 

3.  Partial  prod,  by  4  h'ds  13704       =  4  times  the  multiplicand 
Whole  product  1,565,682  —  457  times  the  multiplicand 


Multiply  647  by  58. 

OPERATION. 
647 

58 


5176 
3235 


37,526  Ans. 


5.     Multiply  28433 
by  4172 


4.     Multiply  21794  by  2365 

OPERATION. 

21794 
2365 

108970 
130764 
65382 

43588 


51,542,810  Ans. 

6.     Multiply  989769 
by  248193 


Multiply  the  following  numbers. 


7.  483  by  569 

8.  924  by  237 

9.  1683  by  328 

10.  581  by  76 


11.  1847  by  84 

12.  2346  by  127 

13.  671  by  508 

14.  8765  by  2043 


Multiplication.  47 

Operation  of  the  13th 
problem. 

671  Explanation. — In  all  problems  where 

508  there    are  naughts  in  the  multiplier 

we  multiply  by  the  significant  figures 

5368  only,  for  the  reason  that  the  product 

3355  of_any  number  by  0  is  0. 

340868 

56.      To  multiply  when  either  the  multiplicand  or  mul- 
tiplier or  both  have  naughts  on  the  rijht. 

1.     Multiply  463  by  200. 

OPERATION.  Explanation. — In  all  problems  of  this 

^gjj  kind  we  write  the  significant  figures  so 

900  *kftt  units  of  the  same  order  may  stand 

in    the    same    column    and    write    the 
naughts  on  the  right  of  the  significant 
92600  figures.     We  then  multiply  the  signifi- 

cant figures  and  annex  to  the  product  as  many  naughts  as  there 
are  in  the  multiplier  or  multiplicand,  or  both.  The  basis  or 
reason  of  this  is  that  the  removal  of  a  figure  or  number  one 
place  to  the  left  increases  its  value  ten  fold,  and  the  annexing 
of  a  naught  removes  the  significant  figures  one  place  to  the  left, 
thereby  increasing  them  ten  fold,  and  hence  annexing  a  naught 
is  in  effect  multiplying  by  10  ;  aid  for  the  same  reason  annexing 
two  naughts  is  multiplying  by  100,  the  annexing  of  3  naughts  is 
multiplying  by  1100,  etc.,  for  other  powers  of  10.  In  this  prob- 
lem we  first  use  the  multiples  2  hundred  as  2  units,  hence  the  first 
partial  product,  926,  was  100  times  too  small,  we  then  by  an- 
nexing the  two  naughts  multiply  it  by  100  and  obtained  92600 
as  the  correct  product. 


Multiply  3400  by  26. 

OPERATION. 
3400 

26 

204 
68 


88400     Ans. 


3.     Multiply  940  by  4700 

OPERATION. 
940 

4700 


658 
376 

4418000  Ana, 


I"          ArUhmetieal  "Exercises  and  Examples. 

Multiply  5020  by  420.  Multiply  S2000  by  4S,'{. 

OPERATION.  S20UO  483 

5020  4s:;  s2ooo 


420 


1004 

200S 


2108400 


210  966 

[356  3864 


39606000 


39606000 


4.  Multiply  842  by  GOO. 

r>.     Multiply  1208  by  1020. 
I).      Multiply  DIMM)  by  707. 

7.    Multiply  2:;:)0o  by  i2o:;o. 

5.  Multiply  1000  by  IJ20S. 

<).      Multiply   SI 00!)  by  <)0200. 

10.  Multiply  45U7H  by  57SO. 

11.  Multiply  !)>7000  by  4!>. 

,">7.       T<*  multiply  ly  the  /'c/r/o/-.s  <tf  u 

NOTK. — FaM'tors  of  a  nuinbiT  are  such  numbers  as  will 
wht3ii  multiplie  I  together  proluoe  the  numhor.  'J'htis  <; 
and  (J  are  the  factors  of  o<>  ;  7  and  8  are  the  factors  of  5f>, 
or  it  is  a  number  that  will  exactly  divide  a  number. 

1.     Multiply  2435  by  42. 

OPERATION. 

•>4;-»f,  Explanation. — In  all  problems  of  this 
1                  kind  we  separate  the  multiplier  into  two 
or  more  factors  and  multiply  the  multi- 
plicand by   one  of  the  factors   and  the 
17045                  resulting  product  by  another  factor  and 
(J                  so  on  until  we  have  used  all  the  factors. 
The  last  product  will  be  the  correct  pro- 
duct. 


102270 
2.     Multiply  781  by  63. 
:•*.     Multiply  3140  by  36. 
4.     Multiply  588  by  81. 


5.  Multiply  480  by  361. 

6.  Multiply  1756  by  125 

7.  Multiply  3281  by  128 


58.    To  multiply  when  the  Multiplicand  or  multiplier  con- 
tains dollars  and  cents. 


Multiplication.  49 

1.  Multiply  $342.15  by  6. 

OPERATION.  Explanation. — In  all  problems  of 

$342  15  *^s  kind  we  multiply  in  the  regu- 

lar manner  and  then  prefix  the 
dollar  sign  $  and  place  the  p<^it 
(.)  two  places  from  the  right.  Wir 

Product     $2052.1)0  answer    is    then    in    dollars    and 

cents. 

EXAMPLES. 

2.  What    will    1082   pounds  of  sugar  cost  at  9/  per 
pound  ?  Ans.  $151.38. 

3.  A  merchant's  monthly  expenses  are  $1342.75.   What 
are  they  for  12  months?.    '  Ans.  $1  Gil 3.00 

4.  It  costs  a  family   $2.30  a  day  for  marketing,   what 
will  be  the  expense  for  30  days  ?  Ans.   861UM). 

5.  What  will  37  boxes  oranges  cost  at  83.75  per  box? 

Ans.  $138.75. 

(>.  At  16  cents  per  pound  what  is  the  value  of  23780 
pounds  Cotton  ?  Ans.  $3804.80. 

7.  If  it  costs  $17500  to  construct  one  mile  of  railroad 
what  would  be  the  cost  to  build  3(J4  miles  .' 

Ans.  sr.:;70ooo. 

8!  What  will  S75  tons  of  railroad  iron  cost,  at  $55  per 
ton  ?  Ans.  £4S125. 

!>.  Multiply  one  million  and  twenty-six  by  nineteen 
thousand  seven  hundred  and  ten.  Ans.  11)710512460. 

10.  One  cubic  foot  contains    1728  cubic  inches.     How 
many  cuijic  inches  in  324  cubic  feet  ?         Ans.  559872. 

11.  One  square  foot  contains  144  square  inches.     How 
many  square  inches  in  !>5  square  feet?          Ans.  13680. 

12.  One  gallon  contains  231  cubic  inches.     How  many 
cubic  inches  in  a  cistern  that  holds  3500  gallons  ? 

Ans.  808500. 

13.  One  bushel  contains  2150.42  cubic  inches.     How 
many  cubic  inches  in  20  bushels  ?  Ans.  43008.40. 

14.  One  mile  contains  5280  feet.     How  many  feet  in 
25  miles?  Ans.  132000. 

• 


r>n         Arithmetical  Exercises  and  Examples. 

15.  One  year  contains  :>r>5  days.  How  many  days  in 
21  years?  Ans.  7665. 

1(>.     The  human  heart  beats  4200  times  an  hour.     How 
many  times  does  it  boat  in  10  years,  there  being  24  hours  in 
one  day  and  :*i;5  day.-  in  each  \var?      Ans.   .'Ji»7i^OUUO. 
™  17.     Sound  travels  11  IS  foot  per  second.     How  far  will 
it  travel  in  10  minutes,  there  boini;  60  seconds  in  a  minute. 

Ans.     670800  feet. 

1 S.     Light  (ravels  1  H2500  miles  per  second.     How  many 
miles  will  it  travel  in  1  day,  there  being  24  hours  in  a  day, 
60  minutes  in  an  hour,  and  GO  seconds  in  a  minute. 
^  Ans.     16,632,000,000. 

It).  A  railroad  train  runs  25  miles  an  hour.  How 
far  will  it  uu  in  :J  days,  allowing  3  hours  for  lost  time  in 
stoppa.L-.  Ans.  1725. 

20.  1.1'  a  person  respire  20  times  in  a  minute,  how  many 
times  will  he  breathe  in  a  day?  Ans.     28,800. 

21.  If  a  person  inhales  1  gallon  of  air  at  each  respira- 
tion, and  respires  20  times  per  minute,   how  many  gallons 
will  he  inhale  in  24  hours.  Ans.   28800. 

22.  At  §17  per  ounce  what  is  the  worth  of  9  pounds 
of  uold,  there  being  12  ounces  in  a  pound  Troy  or   Mint 
weight?  Ans..   $1836. 

23.  How  many  pounds  of  coffee  in  180  bags  if  each 
bag  contains  162  pounds?  Ans.  29160. 

24.  How  many  pounds  of  cotton  in  87  bales,  if  each 
bale  weighs  475  pounds  ?  Ans.     41325. 

25.  What   will   27893   pounds  of  tobacco  cost  at  56 
cents  per  pound  ?  Ans.     $15,620.08. 

26.  What  will  1870  acres  of  land  cost  at  $18  per  acre  ? 

Ans.     $33,660. 

27.  The  Senate  and   House  of  Representatives  of  the 
State  of  Louisiana  consists  of  138  members  who  receive  $8 
per  day.     The  regular  session  continues  60  days.     What 
is  the  yearly   expense  for  the  salaries  of  the  State's  law 
makers?  Ans.     $66240. 

27.  A  contractor  has  865  men  employed  at  $1.50  per 
day.  What  are  the  weekly  wages  of  all  for  6  days'  labor? 

Ans.  $7785. 


Multiplication.  51 

28.  What  will  it  cost  to  build  37428  cubic  yards  of 
levee  at  45  cents  per  cubic  yard  ?          Ans.  $16842.60. 

29.  A  steamboat  arrives  with  3840  bales  of  cotton  ; 
1320  sacks  cotton  seed  and  580  barrels  molasses.     Her 
freight  charges  are  $2  per  bale  for  cotton,  25/  per  sack  far 
cotton  seed  and  50/  per  barrel  for  molasses.     What  is  the 
amount  of  her  freight  bills?  Ans.  $8300. 

30.  A  drayman   charges  75  cents  a  load,  and  he  has 
hauled  63  loads.     How  much  is  due  him  ? 

Ans.  847.25. 

31.  What  will  it  cost  to  slate  the  roof  of  a  house  contain- 
ing 52  squares  at  $13.25  per  square  ?  Ans.  $689. 

32.  The  walks  around  a  dwelling  contain  129  square 
yards.     What  will  it  cost  to  flag  them  with  German  flans  at 
$3.10  per  square  yard?  Ans.  $399.90. 

33.  What  will  it  cost  to  pave  a  street  containing  20000 
square  yards,  with  stone  at  $4.75  per  square  yard  ? 

Ans.  $95000. 

34.  Bought  2180  barrels  of  coal  at  4S/  per  barrel. 
What  was  the  cost?  Ans.  £104ti.4o. 

35.  Multiply   5   billions  and   16  l»y  5  millions   and  1 
thousand.  Ans.  25,005,000,080,016,000. 

36.  A  Hogshead  of  sugar  contains  10S5  pounds;  how 
many  pounds  in  107  hogsheads  of  equal  weight  ? 


37.  A    planter   produced    68    bales  of  cotton,  if   the 
average  weight  of  the  bales  was  460  pounds,  and  the  cotton 
sold  for  13  cents  per  pound,  bow  much  money  would  it 
bring?  Ans.     $4,066.40 

38.  What  will  3  cases  containing  2  dozen  pairs  each  of 
shoes  cost  (a)  $2.90  per  pair  ?  Ans.     §2t»S.sn. 

39.  If  it  costs  $1.50  a  day  to  support  one  person,  what 
will  it  cost  to  support  a  family  of  13  for  one  year  or  .'>(>.") 
days?  Ans.     $7117.50. 

40.  There  are  35600  dwellings  in  New  Orleans,  allow- 
ing 7  persons  to  each  dwelling,  what  would  be  the  popula- 
tion of  the  city  ?  Ans.     249200. 

41.  A  merchant  sold  thrw  dor.cn  dozen  ladies'  hose  at 


52          Arithmetical  Exercises  and  Examples. 


one  quarter  of  a  <1<>\<  n  </<>:,  ,t  cents  a  pair.  How  much 
did  he  receive  fur  them?  A  us.  £155.52 

42.  The  pressure  of  tli  o  atmosphere  is   15   puunds  on 
every   square    inch  nf  surface.      The    exterior    surface  of  a 
m;in  of  average  size  is  aliout   2501}    square    inches.      How 
many  pounds  weight  does  he  sustain  ? 

Ans.     .'J7500  pounds. 

43.  How  many  dollars  aiv  .'>7-")  *lo  o-,,l,l  pieces  worth? 

Ans.     $-J750. 

44.  What  is  the  value  uf  21  -HI  dimes?     Ans.   82  14.  (JO 
-IT).      AYhat  is  the  valucof  1010  ,,uar.  dol.  Ans.   $252.50 
4(>.      What  is  the  value  of    72S  nirkels  ?    Ans.      s.'ili.lO 
-17.     What  is  the  value  of  1612  hall1,  dol.  Ana.  ssou.oo 

48.  During  the  fiscal  year  ending  Sept.  1,  ls7i>,  there 

was  received  .'501  Si  hn^slu-ads  nf  Tuhaeeu.  11'  each  hlid. 
contained  12  pounds  of  poison,  how  many  pounds  of  poison 
were  there  in  the  whole?  Ans.  .".U2172. 

49.  The   circumfer'-nce   of  the   earth    is   nearly   25000 
miles,  the  distance  to  the  sun  is  .'WOO  times  as  many  miles. 
How  far  is  it  to  the  sun  ?  Ans.  95000060. 

50.  -1S75  is  the  thirteenth  part  of  a  number.      What  is 
the  number?  Ans.  (l!5!»7r>. 

51.  The  sun  is   loSl.^oo   nines   as  large  aft  the  earth; 
the  earth    is  4f>    times  as  larire  as  the   moon.      How   many 
times  is  the  sun  larger  than  the  moon  ?    Ans.  62302500. 

___  52.  Lio-ht  travels  l!)2.")00  miles  a  second  and  it  requires 
100000  years  to  travel  to  us  from  some  of  the  fixed  stars 
that  are  seen  with  the  telescope.  Allowing  !>(>.')  days,  5 
hours,  48  minutes  and  41)  seconds  to  a  year  and  remembering 
that  there  are  24  hours  in  a  day,  (JO  minutes  in  an  hour 
and  60  seconds  in  a  minute,  how  far  distant  arc  such  stars? 
Ans.  607470883250000000  miles. 

53.  A  man's  receipts  are  $1800  a  year  and  his  disburse- 
ments are  §1125  a  year.     How  much  arc  his  net   receipts 
in  3  years?  Ans.  §2025. 

54.  It  is  estimated  by  Astronomers  that  7500000  vis- 
ible meteors  fall  upon  the  earth  daily  ;  it  is  also  estimated 
that  the  average  weight  of  each  is  100  grains.     From  thcs<- 


Division — Decreasing.  53 

figures  and  allowing  365  days  to  the  year,  what  is  the  annual 
growth  of  the  earth  in  weight  by  the  accession  of  the  visible 
meteoric  matter  ?  Ans.  273750000000  grains. 

DIVISION.— (Decreasing.) 

59.  Division  is  the  process  of  finding  how  many  times 
one  number  is  equal  to  another.    Or  it  is  the  process  of  find- 
ing one  of  the  factors  of  a  given  product  when  the  other 
factor  is  known. 

60.  The  Dividend  is  the  number  to  be  divided  or  it  is 
the  number  to  be  measured. 

61.  The  Divisor  is  the  number  by  which  we  divide  or 
it  is  the  number  used  as  a  unit  of  measure. 

62.  The  Quotient  is  the  result  of  the  division,  and 
shows  how  many  times  the  dividend  is  equal  to  the  divisor. 

63.  The  Remainder  is  the  number  left  after  dividing 
dividends,  which  are  not  multiples  of  the  divisor,  or  which 
are  not  an  exact  number  of  times  equal  to  the  divisor.     It 

*  must  always  be  less  than  the  divisor. 

64.  The  Sign  Of  Division  is  a  horizontal  line  with  a 
point  above  and  below,  thus  -f-.     It  is  read  divided  l>y  ;  and 
it  indicates  that  the  number  before  it,  is  to  be  divided  by  the 
number  after  it ;  thus  25  -j-  5,  is  read  25  divided  by  5. 

The  horizontal  line,  and  the  vertical  or  curved  line  when 
placed  between  two  numbers  also  indicates  division.  Thus, 
-3j6,  4|36  or  4)36,  are  all  read  36  divided  by  4. 

65.       PRINCIPLES   OF   DIVISION. 

1.  When  the  divisor  and  dividend  are  both  denominate 
or  both  abstract   numbers,  the  quotient  will  be  an  abstract 
number. 

2.  When  the   divisor  is  an  abstract  number  and  the 
dividend   a   denominate  number,   the  quotient  will   be  a 
denominate  number. 

3.  When  there  is  a  remainder  it  is  a  part  of  the  divi- 
dend and,  is  therefore  the  same  in  name  or  kind. 

B* 


54          Arithmetical  Exercises  and  Examples. 


4.  Multiplying  the  dividend  or  dividing   the  divisor 
i))nltij)/irs  the  quotient. 

5.  Dividing  the   dividend  or  multiplying    the   divisor 
divides  the  quotient. 

(5.  Multiplying  or  dividing  both  the  divisor  and  divi- 
dend by  the  same  number  does  not  change  the  quotient. 

66.  Prnnf  of  Division.  Multiply  the  quotient  by  the 
divisor  and  if  there  is  no  remainder  the  product  should  he 
equal  to  the  dividend  ;  when  there  is  a  remainder  add  it  to 
the  product,  and  it'  the  work  is  correct  the  sum  will  equal 
the  dividend. 

I  >i  vision  operations  may  be  performed  by  the  process 
of  addition  or  subtraction.  Hut  as  tlies  •  processes  are  too 
lengthy  for  practical  purposes,  we  will  not  give  them  place 
here. 

For  contracted  methods  in  division,  see  Sonle's  Con- 
tractions in  Numbers. 


1. 


3. 
4. 
5. 
6. 

7, 
8. 
9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17, 


67.       ORAL  EXERCISES. 
How  many  times  is   0  equal  to   1  ?  or    0 —  1 
"  "  1        u         0?or    1 

Ans.  An  infinite  number  of  times. 

How    many    times  is  1  equal  to  1  ?  or    1 

2       "         1  ?  or    2 

"  3       "         1  ?  or    3 

"  "  4       "          2  ?  or    4- 

"  "  8       "         2  ?  or    8- 

"  "  !l       "         :*?  or    9 

"  "  12       u         4?  or  12 

"  «  20       "         5?  or  20- 

"  24       "          6  ?  or  24 

-35       "         7  ?  or  35 

"  "  56        "         8  ?  or  56- 

«  "  63  9  ?  or  63 


72 
80 
88 
96 


9  ?  or  72 
10  ?  or  SO- 
11  ?  or  88- 
12  ?  or  96 


=  ? 


Division — Fractional  Numbers.  55 

19.  3g6-=?         4)42=?         9)45=?         77-=-  7? 
-%*=?         6)48=?         5)55=?         84—12? 

20.  How  many  times  is  24  equal  to  3,  to  4,  to  6,  to  8, 
to  12,  to  24? 

21.  How  many  times   is  36  equal  to  3,  to  4,  to  6,  to  9, 
to  12,  to  36? 

22.  How  many  times  is  42  equal  to  2,  to  6,  to  7,  to  42  ? 

23.  "  "          64       "        2,  to  4,  to  8,  to  64  ? 

24.  "  "          72      "        2,  to  8,  to  9,  to  72  ? 

68.       FRACTIONAL    NUMBERS. 

When  we  divide  a  unit  or  a  number  of  units  of  any 
kind  into  equal  parts,  these  parts  are  sometimes  called  frac- 
tions. The  name  of  the  equal  parts  varies  according  to 
the  number  of  parts  into  which  the  thing  or  number  was 
divided. 

When  the  unit  or  number  is  divided  into  2  equal  parts 
1  of  the  parts  is  called  om--h<i1j\  and  is  written  thus  J.  If 
divided  into  4  equal  parts  1  of  the  parts  is  called  one-fourth, 
and  is  written  thus  \  ;  3  of  the  parts  are  called  three-fourths 
and  are  written  thus  j. 

In  like  manner  we  obtain  fifth  x,  .s/./-/Ax.  *  ?•/  ,/fhs,  eighths, 
twelfths^  sixteenths,  twenty-firsts^  etc. 

In  writing  fractional  numbers  in  figures  we  place  the 
number  which  shows  the  name  of  the  parts  below  a  hori- 
zontal line  as  a  divisor,  and  the  number  which  shows  lum: 
many  parts  are  taken  or  used,  above  the  line  as  a  dividend. 

The  following  examples  will  fully  elucidate  this  work  : 


Two-thirds  are  written  |. 
Three-fourths  are  written   ,: 
Five-eighths  are  written  § 
Seven-ninths  are  written  /. 


Seven-twelfths  are  written 
Nine-tenths  are  written  ,'',, 
Fifteen-sixteenths  written 
Eleven-eightieths  written 


How  do  you  find  •],  .1,  1 ,  ^,  etc.  of  any  number? 
How  do  you  finu  .  A,  etc.  of  any  number? 


What  is  \  uf  4? 
"  "  J  of  <J? 
"  «  1  of  8? 


What  is  4- of  15? 
"  "  i  of  18? 
"  "  \  of  28? 


What  is  5  of  !)? 
of  12? 
of  40? 


7 


5(>          Arithmetical  Exercises  and  Examples. 

EXAMPLES. 

1 .     If  3  hats  cost  8l>  what  will  1  hat  cost  ? 

.      Ans.   £2. 

Analytic  solution. — Since  3  hats  cost  $0,  1  hat  will  cost  J  part 
of  S»J,  which  i« 

L>.     If  8  yards  cost  56  cents  what  will  1  yard  cost  ? 

:>.  Paid  *:><»  for  (>  barn-Is  of  flour,  what  did  1  bam-1 
cost  ? 

4.  0  gallons  of  molasses  cost  §4.50,  what  did  1  gallon 
cost?  AHS.  50/. 

5.  Bought  12  shirts  for  $30,  how  much  did  1  cost? 

Ans.  $2.50. 

(>.  Paid  SI. 00  for  8  pounds  of  sugar,  what  was  the 
price  per  pound  ?  Ans.  &.12]. 

7.  7  dozen  oranges  cost  $2.10,  what  was  the  price  per 
dozen?  Ans.  8. .'Jo. 

8.  Bought  20  peaches  for  60^,  how  much  did  1  peach 
cost?  Ans.  $.03. 

9.  At  $2  a  yard  how  many  yards  can  you  buy  for  $'1 1  ? 

Ans.   12  yards. 

Analytic  solution. — Since  $2  buy  1  yard,  $1  will  buy  .]  of  a 
yard,  and  $24  will  buy  24  times  J  a  yard,  which  is,  y  or  12 
yards. 

Or  thus.  Since  $2  buy  1  yard,  for  $24  we  can  buy  as  many 
yards  as  $24  is  equal  to  $2,  which  is  12  times. 

10.  At  9  cents   per  pound  how  many  pounds  can  be 
bought  for  45  cents  ?  Ans.  5  pounds. 

Analytic  solution. — Since  9  cents  buy  1  pound,  1  cent  will  buy 
^  of  a  pound,  and  45  cents  will  buy  45  times  ^  of  a  pound, 
which  are  -\5-  or  5  pounds. 

11.  Flour  is  worth  §8  per  barrel,  how  many  barrels  can 
be  purchased  for  $56  ?  Ans.  7  barrels. 

Analytic  solution. — Since  $8  buy  1  barrel,  $1  will  buy  J  of  a 
barrel,  and  $56  will  buy  56  times  J  of  a  barrel,  which  are  -%6  or 
7  barrels. 

12.  For  $,95  how  many  papers  can  you  buy  at  5  cents 
a  paper?  Ans.  19  papers. 


Short  Division.  57 

13.  If  25  cents  buy  1  yard   how  many  yards  will  75 
cents  buy  ?  Ans.  3  yards. 

14.  At  $3  a  piece  how  many  chairs  can  be  bought  for 
$36?  Ans.   12  chairs. 

15.  If  the  printer  charges  §1.50  to  set  1  page  of  this 
book,  how  many  pages  can  be  set  for  $75  ? 

Ans.  50  pages. 

WRITTEN    EXERCISES. 
69.      To  divide  irJten  the  divisor  docs  not  exceed  12. 

1.  Divide  3048  by  5. 

OPERATION.  Kffhniation. — In  all  problems  of 

Divisor  5)  3048  dividend   this  kind  we  write  the  numbers  as 

shown  in  the  operation,  and  then 

begin  on  the  left  of  the   dividend 
Quotient  729  and  3rem.  to  (livide>     We  bc?in  on  the  left 

in  order  to  carry  the  remainder,  it'  any,  of  the  higher  order  of 
units  to  the  next  lower  order.  In  this  problem  we  first  take 
the  3  (thousands,)  and  as  it  is  not  equal  to  5,  we  therefore 
unite  it  with  the  <>  hundreds,  making  36  hundreds,  which  by 
trial  multiplication  and  subtraction  mentally  performed,  we 
find  is  equal  to  5,  7  (hundreds)  times  and  1  remainder;  the  7 
we  write  in  the  hundreds  column  of  the  quotient  line,  directly 
under  the  »>  the  last  figure  used  of  the  dividend  ;  then  to  the 
1  remainder  we  mentally  annex  the  -1  tens,  making  14  tens,  as 
the  second  partial  dividend,  and  which  by  mental  multiplication 
and  subtraction,  we  find  is  equal  to  5,  2  (tens)  times  and  4 
remainder;  the  2  we  write  in  the  tens  column  of  the  quotient 
line,  and  to  the  4  we  mentally  annex  the  units  figure  of  the 
dividend,  making  48  units  as  the  third  and  last  partial  dividend  ; 
this  we  find,  by  mental  multiplication  and  subtraction  to  be 
equal  to  i">,  !>  times  and  3  remainder. 

The  remainder  is  usually  expressed  fractionally  by  writing 
it  over  the  divisor,  thus  2,  this  expresses  the  part  of  a  unit  of 
times  that  the  remainder  is  equal  to  the  divisor. 

SHORT   DIVISION". 

Operations  in  division  according  to  the  foregoing  method 
are  called  short  divixittn,  because  tlui  multiplication  and 
subtraction  work  in  finding  the  remainder  of  the  partial 
dividends  were  mentally  performed. 

2.  II ow  many  times  is  840  e<jual  to  0  ? 


58          Arithmetical  Exercises  and  Example*. 


OPERATION. 
Divisor  0)846    Dividend 


Quotient     141 


tinn. — In  the*  pre- 
cxMling  problem  we  gave  a 
full  and  explicit  explanation 
of  each  step  of  the  operation. 


In  practice  much  of  the   explanation  therein  given  is  omitted, 
and  the  work  performed  thus  :  Commencing  with  the  left  hand 
figure  we  say  8  is  equal  to  0,  1   time  and  2  remainder;   24,   is 
equal  to  6,  4  times  ;  G  is  equal  to  6,  1  time. 
Work  the  following  indicated  divisions. 

7)847          8)12327          9)1  n<»          ll)2:;-i; 


121 
8)1471 

1540J 

4)1  1 

120| 
9)81018 

n 
2344 

7  1  93020 

21345--8 

12)10824 


9  76451 


Divide  the  following  numbers  : 

15.  9872  by  4 

16.  1483    "   7 

17.  1691    "   9 

18.  41070   "  8 

23.  What  is  i  of  $528?   1 

24.  u   aref  of  §1005?  \ 

Operation  for  the  24th 

problem. 
5)$1005 


3 


$603    Ans. 


19.  10286  by  6 

20.  48710   "  7 

21.  1001)8   "   9 

22.  199999  "   8 
W hut  are  1  of  $448? 

"      "I  of  §6444? 

Operation  for  the  26th 
problem. 

8)$6444 


25. 

26. 


805.50= 

7 


$5638.50    Ans. 


27.     How  many  apples  can  be  bought  for  $  2.25  at  5 
cents  a  piece  ?  Ans.  45  apples. 


Short  Division.  59 

28.  At  15  cents  a  pound,  how  many  pounds  can  you 
buy -for  S3. 15?  Ans.  21  pounds. 

29.  Paid  $90  for  10  volumes  of  Chambers'  Cyclopedia, 
what  was  the  price  of  one  volume  ?  Ans.  $i). 

30.  If  8  men  are  to  receive  $5791  in  equal  parts,  what 
wijl  be  each  man's  share  ?  Ans.  $723-J. 

31.  The  dividend  is  63,  and  the  quotient  is  9,  what  is 
the  divisor  ?  Ans.  7. 

32.  The  quotient  is  15,  the  divisor  3,  and  the  remain- 
der 2,  what  is  the  dividend  ?  Ans.  47. 

33.  The  quotient  is  3G,  and  the  divisor  6,  what  is  the 
dividend?  Ans.  216. 

34.  The  dividend  is  72,  and  the  divisor  is  4,  what  is 
the  quotient?  Ans.  18. 

35.  "/l^ow  many  pounds  of  cotton  at  11  cents  a  pound 
will  be  required  to  pay  for  33  pounds  of  sugar  @  8  cents 
a  pound  ?  Ans.  24. 

70.      To  divide  when  the  divisor  exceeds  1 .. 
1.     Divide  7387  by  36. 

OPERATION.  Kjcjtlnnation. — We  first  write  the 

Divisor,  Dividend,  Quotient  numbers  as  shown  in  the  opera- 

•->r\          '7'-*Q-'       /»>nr  7     tion  and  commence  to  divide  as 

fttU  ^Vo    explained  in  the  first  written  ex- 

'-•  ample.     But  as  the  divisor  is  too 

large  to  be  conveniently  used  men- 

1  S7  tally,  we  therefore  write  tho  oper- 

jgQ  ation  of  multiplying  the   divisor 

by  the  quotient  figures,  and  sub- 

^  .    ,       ttacting  the  successive  products 

7  remainder  from  tne  several  partial  dividends. 
In  performing  the  division  we  first  see  that  7,  (thousands) 
are  not  equal  to  36,  and  hence  there  will  be  no  thousands  in  the 
quotient.  We  then  annex  to  the  7  thousands  the  3  hundreds, 
making  73  hundreds  as  the  first  partial  dividend;  this  is  equal 
to  30,  2  times,  and  a  remainder;  we  write  the  2  in  the  hundreds 
column  of  the  quotient,  multiply  the  divisor  by  it,  write  the 
product  under  and  subtract  the  same  from  the  73  hundreds  of 
the  dividend.  This  work  gives  us  1  hundred  remainder,  to 
which  we  annex  the  8  tens,  making  18  tens  as  the  second  partial 
dividend  ;  this  partial  dividend  not  being  equal  to  36,  we  write 


00          Arithmetical  Exercises  and  Examples. 


0  (no  tens)  in  the  tens  column  of  the  quotient,  ami  annex  to 
the  18  tons  the  7  units,  making  187  units  as  the  third  and  last 
partial  dividend.  This  is  equal  to  36,  5  times  and  a  remainder, 
\ve  write  the  5  in  the  quotient,  and  multiply  and  subtract  as  we 
did  with  the  first  obtained  figure  of  the  quotient,  and  thus  pro- 
duce 7  remainder,  which  we  write  over  the  divisor  as  explained 
in  short  division. 

LONG   DIVISION. 

Operations  in  division,  according  to  the  above  method, 
are  called  long  tl  iris  fun,  for  the  reason  that  the  multiplica- 
tion and  subtraction  work  in  finding  the  remainders  of  the 
partial  dividends  is  written. 

2.     How  many  times  is  66804  equal  to  5'1  ? 


OPERATION 

Divisor   Dividend 

53)  i;r,sn4  0-<; 
53 

138 
106 


Proof. 

1260  Quotient, 

53  Divisor. 


3780 
6300 

24  Remainder. 


320 
318 

24 

3.     What  is  the  quotient 
of  107941 --396? 

OPERATION. 

396)107941(272  Quotient. 
792 


2874 
2772 

1021 
792 


229  Remainder. 


66804  Dividend. 


4.     Divide     7167901     by 
11 207. 


OPERATION. 


11267)7167901(636  Quoti't. 
67602 


40770 
33801 

69691 
67602 

2089  Remainder. 


Long  Division.  61 


5.     Divide  784  by  82. 

STATEMENT. 

82)784(9|f  Ans. 


6.     Divide  91070  by  8761. 


STATEMENT. 


8761)91070(10-ff£f  Ans. 


7.  Divide  2461  by  74. 

8.  Divide  4809  by  91. 

9.  Divide  13872  by  263. 

10.  Divide  54123  by  1423. 

11.  Divide  628100  by  156. 

12.  Divide  10000  by  304. 

13.  Divide  37021  by  2002. 

4    Divide  8888888  by  332211. 
Divide  $6805  equally  between  5  men,  and  what 
will  be  the  share  of  each  ?  Ans.  §1361. 

16.  What  is  the  sir.iy-fniirth  part  of  $44800? 

Ans.  $700. 

17.  145  men  picked   1305000  pounds  of  cotton,  sup- 
posing they  all   picked   an  equal   quantity,  how  much  did 
one  man  pick  ?  Ans.   9000  pounds. 

18.  A  father  gave  his  7  sons  a  Christmas  present  of 
1353.50  to  l>c  shared  equally,  what  was  each  one's  share? 

Aus.  $50.50. 


71.       To  divide  w/trn    th<*i-<    «r>    nnnylitu  on  the  right  of 
tin1  divisor. 

1.     Divide  2843  by  200.  Ans.  14^fr 

OPERATION.  Explanation.  —  Since  by  our  scale 

°'M))2843(  of   n»mbers    tncy   increase    from 

I      '  ~*(       __  right  to  left  in  a  tenfold  ratio,  and 

~~  decrease   from  left  to  right  in  a 

14  and  4.5  Kern,  corresponding  manner,  it  is  clear 

t.h:it  the  removal  of  any  order  of  figures  from  left  to  right  dimin- 

ishes its  value  ten  times  for  each  place  of  removal.     And  as 

previously  shown,  that  the  annexing  of  naughts  multiplies  num- 

bers, by  removing  them   to  places  of  higher  value,  so  in  like 

manner  cutting  figures  off  from  the  right  of  a  number  removes 

the  remaining   orders  to  the  right,  and  hence   decreases  them 

tenfold  for  every  figure  cut  off.     Hence  to  cut  off  one  figure  is 

dividing  by  10  ;  to  cut  off  two  figures  divides  by  100  ;  to  cut  off 

three  figures  divides  by  1000  and  so  on. 


62 


Arithmetical  Exercises  and  Examples. 


Considering  these  principles,  in  all  cases  of  this  kind  we  cut 
off  the  naughts  from  the  right  of  the  divisor  and  the  same  num- 
ber of  figures  from  the  right  of  the  dividend  ;  and  then  divide 
the  remaining  figures  of  the  dividend  by  the  remaining  figun-s 
of  the  divisor.  When  there  is  a  remainder  annex  the  figures 
nit,  off,  and  we  obtain  the  true  remainder. 

2.  Divide  S7JK51  by  1000.  Ans.  87T°0Vu- 

OPERATION. 

1|000)  87(931 

Quotient  87  and  931  Remainder. 

3.  Divide  178  by  10.          I     4.     Divide  6581  by  :;on. 

OPERATION.  OPERATION. 

10;17S  3 00)65 H 

Quotient  21— 2S1    Rein. 
Ans.  LM 


Quotient  17 — 8  Remainder 
Ans.  17 /V 


6. 


Divide  714G8071  by'.341000. 

OPERATION. 


Ans. 


682 

3268 
3069 


6. 
7. 
8. 
9. 
10. 


199 

Divide  8897600  by  8100. 
Divide  1000000  "  10000. 
Divide  99999  by  9000. 
Divide  33440  by  270. 
Divide  140817  by  6800. 


Ans.  1098-ff. 
Ans.  100. 
Ans.  ll-fffo. 
Ans.  123ffg, 
Ans.  20f  ffrf 


Division.  63 

72.      To  divide  by  the  Factors  of  a  number. 

1.  Divide  936  by  24.  Explanation.— In  ail  problems 
OPERATION.                        where  the  divisor  is  a  compos- 

4)936  ite  number  we  may  divide  by 

the  factors  and  thus  shorten  the 

operation     In  this  example  the 
factors  are  4  and  6,  and  we  first 
divide  by  4  which  gives  a  quo- 
39  tient  6  times  too  large,  for  the 

reason  that  4  is  but  J  of  24  the  true  divisor.     We  therefore  divide 
this  quotient  by  6  and  obtain  the  true  quotient. 

2.  Divide  588  by  28.     The  factors  are  4  and  7. 

ADS.  21. 

3.  Divide  6976  by  32.     The  factors  are  4  and  8. 

Ans.  218. 

4.  Divide  2583  by  63.     The  factors  are  7  and  9. 

Ans.  41. 

5.  Divide  10206  by  81.     The  factors  are  9  and  9. 

Ans.  126. 

6.  Divide  11984  by  56.     The  factors  are  8  and  7. 

Ans.  214. 

7.  Divide^  1607  by  72,  using  the  factors  3,  4  and  6,  and 
find  the  true 'remainder. 

Ans.  22  quotient,  and  23  remainder. 

FIRST   OPERATION. 
3^)  1607  Explanation — In  this  ex- 

'  ample  using  as  divisors  3, 

.  4  and  6,  the  factors  of  72. 

4)535          2,  1st  remainder,    we  obtain  for  remainders  2, 

3  and  1. 

6)  133          3,  2d  remainder.       The  first  remainder  2,  is 

clearly   units  of  the  given 

9»>  i     Qrl  *    v,  ;    1         dividend,  and  hence  a  part 

1,  3d  remainder.    of  the  tr'ue  remainder/ 

The  second  remainder,  3  being  fourths  of  the  second  dividend. 
535  which  are  reciprocal  thirds  of  the  given  dividend,  it  is  hence 
|  of  the  reciprocal  of  J  of  J  —  9,  of  the  given  dividend  and 
true  remainder.  41 

The  third  remainder,  1  being  sixths  of  the  third  dividend  133, 
which  are  reciprocal  twelfths  of  the  given  dividend,  it  is  hence 
J  of  the  reciprocal  of  J  of  ^  of  ^  —  12  of  the  given  dividend 


G4          Arithmetical  Exercises  and  Examples. 

and  true  remainder.  Therefore  2,  the  first  remainder,  plus  0, 
the  unit  value  of  the  second  remainder,  plus  12,  the  unit  value 
of  the  third  remainder  =_  23,  the  true  remainder.  Or  we  may 
obtain  the  true  remainder  without  considering  the  reciprocal 

relationship  of  the  quotients  ami  divisors,  tlni>  : 

First  remainder.'  •_' 

Plus  2d,  remainder  3.  X  tnc  preceding  divisor  3,  =  !• 

Plus  3d,  remainder  1,  >x  ftU  tlu'  preceding  divisors,  4  and  3=  12 

which  added  gives  the  true  remainder  23 

From  the  foregoing  we  see  that  the  true  remainder  may  be 
obtained  by  adding  to  the  first  remainder  the  product  of  the 
other  remainders  by  all  the  divisors  preceding  the  one  which 
produced  it. 

8.  Divide  7851  by  <i4.  u>ini:  tlu>  factors  S  and  S. 

Ans.   l'2'2  quoti.  -nt.  4i>  remainder. 

OPERATION.  Ks/'ftimrfinn.  —  Here  the   1st  re- 

8)7851  mainder  is  3,  to  which  we  add  the 

'  _  product   of  the   2d  remainder   f>, 

.  multiplied  by  the  preceding  divi- 

—3,  1st;  remainder.  sor  8j  equals  40,  making  43,  the 

true  remainder. 
122—5,  2d  remainder. 

9.  Divide  17803  by  96,  using  the  factor*  2,  3,  4  and  4. 

Ans.  IK, 

OPERATION.  Explanation. 

2)17803 
' 


__  1st  remainder 

3)8901  —  1  2d  remainder  3X3X2—  18 

4^2967  —  0  3d  remainder  IX^X^X^^=        24 

4)741—3 

True  Remainder  43 

'  185—1 

10.  Divide  27865  by  the  factors  of  81. 

Ans.  344  g\. 

11.  Divide  101041  by  the  factors  of  84. 

Ans.   1202|| 


Division.  65 


12.  Divide  899  by  the  factors  of  108.     Ans. 

13.  If  $4691   are   divided  equally  between   35    men, 
what  will  each  one  receive  ?  Ans.  $134^. 

14.  There  are   32   quarts   in   one   bushel,   how  many 
bushels  are  there  in  1536  quarts  ?         Aus.  48  bushels. 

15.  A  hogshead  of  wine  contains   63   gallons.     How 
many  hogsheads  in  2898  gallons  ?     Ans.  46  hogsheads. 

16.  One  of  the  factors  of  10800  is  225  ;  what  is  the 
other?  Ans.  48. 

17.  What  number  multiplied  by  137  will  give  959137 
fbrjhe_product  ?  Ans.  7001. 
""18.     Multiplying   372   by  an    unknown   number  gives 
44640  ;  what  is  the  number?  Ans.  120. 

19.  What  is  the  quotient  of  9126  divided  by  9  ? 

Ans.  1014. 

20.  Divide  four  million,  eight  thousand  and  sixteen  by 
MMDCXLIV.  Ans.  ISlSfffl 

21.  What  number  is  that  to  which,  if  sixteen  be  added 
the  sum  multiplied  by  8  and  13  substracted   from  the  pro- 
duct the  remainder  will  be  33!)  ?  Ans.  2S. 

22.  Theie  is  a  number  from  which  if  you  subtract  55, 
and  divide  the  remainder  by  12,  your  quotient  will  be  36. 
What  is  that  number?  Ans.  487. 

23.  A  merchant  owes  a  debt  of  £1  <S7.">,  which  he  agreed 
to  pay  by  weekly   installments  ot  825.      He   has   made  55 
pavments,  how  many  more  payments  has  he  to  make. 

Ans.  20. 

24.  A  merchant  bought  350  barrels  of  flour  at  $6  a 
barrel,  and  sold  it  at  $7.50  per  barrel.     The  gain  he  gave 
in  equal  parts  to  4  worthy  boys  to  aid  them  in  obtaining  an 
education.     What  was  the  cost  and   selling  price  of  the 
flour,  and  how  much  money  did  each  boy  receive. 

Ans.  82100  cost,  £2625  selling  price,«$131.25 
each  boy  received. 

25.  An  acre  contains  160  square  rods  ;  how  many  acres 
in  a  plantation  containing  123200  square  rods? 

Ans.  770  acres^ 


66         Arithmetical  Exercises  and  Examples. 

2('>.  A  boy  sold  50  oranges  at  5^  each  and  thereby 
gained  $1.50.  At  what  rate  did  he  buy  the  oranges? 

A  i is.  2/  a  piece. 
27.      How  many  times  13G  will  produce  1708? 

Ans.  i:^. 

IN.  Dividejhe  product  of  750  and  875  by  their  differ- 
ence. Ans.  525o. 

21).  The  diameter  of  the  earth  at  the  equator  is  71)25 
miles ;  how  long  would  it  take  a  locomotive  to  travel  that 
distance  at  the  rate  »>f  25  miles  an  hour? 

Ans.  317  hours— 13  days  5  hours. 

30.  The   first    Atlantic   Telegraph    Cable   as   originally 
made  cost  $125^250.      10  miles  of  deep  sea  cable  was  made 
at  a  cost  of  81450   jH>r  mile,   and  25    miles   of  shore   ends 
was   made   at  a.  cost    of  $1250   per    mile.      The    remainder 
cost  $485  per  mile.      How  many  miles  of  Cable  were  made  ? 
^  Ans.   25i 55  mile,-. 

31.  A  grocer  wishes  to  put  3335  pounds  of  sugar  in  3 
kinds  "of  boxes,    containing    respectively   20,   50    and    75 
pounds,  using  the  same  number  of  boxes  of  each  kind  or 
size.      How  many  boxes  will  he  require  ? 

Ans.  23  of  each  size. 

32.  The  Northern  Pacific  Railroad  from  Lake  Superior 
to  Puget  Sound,  as  located,  is  2000  miles  long.     The  esti- 
mated co*t  and  equipment  of  the  road,   including  interest 
i-  > ^5277000.     What  will  be  the  average  cost  per  mile  ? 

Ans.  $42638.50. 

33.  It  is  estimated  that,  by  reason  of  intemperance  the 
United    States   loses    annually    $08400000.     How    many 
School  Houses  costing  $5000  each,  and  how  many  Libraries 
costing    $3000  could  be   established  with  this  amount  of 
money?  Ans.  12300  of  each. 

34.  ^Ten  freedmen  agreed  to  pick  20000  pounds  of  cot- 
ton and  receive  for  their   labor  I  of  the  cotton  picked. 
After  they  had   picked    7000   pounds  4  freedmen    quit, 
leaving  the  other  6  to  finish  the  work.     How  much  cotton 
i>  each  entitled  to  when  the  work  is  finished  ? 

Ans.   140  pounds  each  for  those  who  left,  ami 
573f  each  for  those  who  remained. 


Divison.  .  67 


35.  A  merchant  bought  800  gallons  of  molasses  at 
and  sold  J  of  it  at  72^  a  gallon.     From  the  profit  he 
bought  his  children  a  set  of  Cutter's  Anatomical  and  Phy- 
siological charts,  and  had  $8.20  left.     What  did  the  charts 
cost?  Ans.  $19.80 

36.  The  capacity  of  steam  engines  is  measured  by  horse 
power  ;  and  1  horse  power  is  a  force  that  will  raise  33000 
pounds  1  foot  in  1  minute.     How  many  horse  power  has  a 
steam  engine  that  possesses  a  capacity  of  1188000  pounds? 

Ans.  36. 

37.  The  average  weight  of  man  is  150  pounds.     About 
\  of  this  weight  is  blood.     Allowing  that  the  heart  throws 
out  2  ounces  of  blood  at  each  pulsation,  and  that  it  beats 
72  times  a  minute,  and  that  16  ounces  make  a  pound,  how 
long  will  it  take  the  heart  to  circulate  all  the  blood  in  the 
body  ?  Ans.  3|^  minutes. 

38.  Prof.    Wilson,   a  physician   and  physiologist,  has 
counted  in  the  skin  of  the  palm  of  the  hand  3528  perspi- 
ratory pores  to  [the  square  inch  ;  but  as  there  are  less  to 
the  square  inch  on  some  other  parts   of  the  body,  he  esti- 
mates that  2800  is  a  fair  average  to  allow  to  the  square  inch 
for  the  whole  surface  of  the  body.     The  average  size  man 
has  2  ")()()  5<iuare  inches  of  body  surface,   which  would  give 
7000000  perspiratory  pores.     Through  these  pores  fully  2 
pounds  of  perspiration,  water,  refuse  matter  and  worn  out 
tissue  pass  every  24  hours.     If  a  man  weighs  150  pounds, 
how  long  will  it  take  for  matter  equal  to  the  weight  of  the 
body  to  pass  through  the   perspiratory  pores,  if  they  are 
kept  open  as  they  should  be  by  daily  bathing  ? 

Ans.  75  days. 

:'/.).     Our  earth  is  about  95000000  miles   from  the  sun, 
Neptune,  the  most  distant  member  of  our  Solar  System,  is  1 
about  2850000000.     How   many  times  farther  from   the 
Sun  is  Neptune  than  our  earth  ?  Ans.  30. 

40.  The  velocity  of  the  earth  on  its  yearly  voyage 
around  the  Sun  is  99733  feet  per  second.  The  velocity  of 
a  cannon,  ball  fired  from  a  gun  with  an  average  charge  of 


^          Arithmetical  Exercises  and  Examples. 

powder  is  1750  feet  a  second.  How  many  times  fatter  is 
the  velocity  of  the  earth  than  a  cannon  ball? 

Ans.   :»<;; 

41.     Gco.  Peabody,  of  Ma-  while  living,  to  '21 

schools  and  colleges,  library  associations,  benevolent  socie- 
ties, state  public  schools,  etc.,  not  including  many  private 
presents,  $7S7.">000.  Of  this  an:uunt  $3300000  \\vrc  -ivm 
to  the  public  schools  of  the  South.  What  part  of  the 
whole  spc»-iliid  donation  did  he  gives  to  the  South  ? 

v...      8800000        ' 
Ans-  b       ;;i.v 

4'2.  Stephen  (Jinml,  of  Philadelphia,  gave  SI;O<MMHIO 
for  the  founding  and  support  of  Girard  College.  Soule's 
College  in  New  Orleans  is  worth  £.'iO,000.  How  many 
such  colleges  could  be  built  with  the  amount  of  money 
-iven  by  Mr.  tlirard  to  establish  one  collc-_ 

Ans.   200. 

43.  The  air  which    surrounds   our  earth,   and  of  which 
we  eaih  inhale  (500  gallons  every  hour,  is  composed  of  four 
parts  of  Nitrogen  and  1  pait  of  ( )xygen.      Il<>w  many  gal- 
lons of  each   are  iherc  in  a  room  2'2  feet   long   and  21  leet 
wide  and  10  feet  high,   which   contains  IM^UO   gallons  of 
air?  Ans.    (JIU2  Oxygen.  L'Tn'iS  Nitrogen. 

44.  A  room  contains  iM-^liO  gallons  of  air,  a  man  inhales 
u'OO  gallons  per  hour,  how  long  will  it  take  for  10  men   to 
inhale  the  air  in  the  room.  Ans.  .")(|;";|;;;-  hours. 

4,").  A  room  1(5  feet  long,  10  feet  wide,  and  8  feet  high, 
contains  12SO  cubic  feet  of  air.  Every  time  a  person 
breathes  he  throws  out  from  his  lungs  a  sufficient  quantity 
of  carbonic  acid,  or  carbon  di-oxide,  (a  most  deadly  gas.) 
to  pollute  or  render  poisonous  and  unfit  for  breathing  3 
cubic  feet  of  air,  and  he  breathes  '20  times  a  minute.  How 
long-  will  it  take  for  the  air  of  a  room  of  the  above  dimen- 
sions to  become  poisonous  if  occupied  by  5  persons,  and  DO 
change  of  air  is  made  by  ventilation. 

Ans.  4-3%°-^  minutes. 

46.  A  man  produces  by  breathing  at  least  6  gallons  of 
carbonic  acid  gas  every  minute,  a  single  burning  gas  jet,  10 
gallons,  an  ordinary  stove,  60  gallons,  How  many  gallons 


Division.  69 

of  carbonic  acid  gas  will  an  audience  of  1000  people,  2  heated 
stoves,  and  50  burning  gas  jets  produce  in  3  hours,  and 
how  many  times  would  the  quantity  fill  a  room  100  feet  long, 
50  feet  wide  and  30  feet  high  ? 

Ans.  1191600  gallons.     l^WoVA  time- 

NOTE. — Theie  are  60  minutes  in  an  hour,  231  cubic  inches  in 
a  gallon,  and  1728  cubic  inches  in  a  cubic  foot. 

47.  Astronomers  estimate  that  7500000  visible  meteors 
fall  upon  the  earth  daily,  the  average  weight  of  which  is 
estimated  to  to  100  grains.  Allowing  for  an  equal  quan- 
tity of  matter  to  be  brought  down  by  the  invisible  meteors 
and  the  serolhes,  how  many  pounds  a  year  does  our  earth 
increase  in  weight,  there  being  7000  grains  in  a  pound,  and 
365  days  in  a  year?  Ans.  78214285f  pounds. 

73.       PROBLEMS     INVOLVING    THE     ENGLISH     MONEY    OF 
ACCOUNT. 

1.  What  will  13840  pounds  of  cotton  cost  at  8  pence 
a  pound.  Ans.  £461.  6s.  8d. 

OPERATION.  Estimation. — In    this    problem    the 

13840  price  is  given  in  one  of  the  subdivi- 

o  *  sions  of  the  English  monetary  unit, 

and  hence  we  must  know  what  that 

unit  and  its  subdivisions  are,  before 

12)  110720d.  we  can  solve  the  problem.     The  En- 

glish  monetary  unit  is  the  Pound  Ster- 

20)  9226      8d       liuff,  which  is  divided  into  20  Shillings ; 
each  shilling  is  divided  into  12  Pen- 
"  nies,  and  each  penny  into  4  Farthings. 

uS-  With  this  knowledge  of  English  money 
we  can  work  all  problems  of  the  above  character.  In  this 
example  we  first  multiply  the  price  of  one  pound  by  the  num- 
ber of  pounds  and  thus  produce  the  value  of  the  whole  in 
pence.  Then  to  reduce  the  pence  to  shillings,  we  divide  them 
by  12,  and  obtain  9*22(5  shillings  and  a  remainder  of  8,  which 
being  a  part  of  the  dividend  is  therefore  8d.  Then  to  reduce 
the  shillings  to  pounds  we  divide  them  by  20,  and  obtain  4GI 
pounds  and  a  remainder  of  6,  which  being  a  part  of  the  second 
dividend  is  therefore  6s.  N  the  English  monetary  system  the 
following  abbreviations  are  used  :  £.  represents  pounds,  s. 
represents  shillings,  d.  represents  pence,  and  f.  represents  far- 
things. 


70         Arithmetical  Exercises  and  Examples. 

'1.     What  is  the  value  of      3.      What  will   241    boxee 
483  yards  of  cloth  at  1(J  shil-  cheese  cost  at  £3  per  box  ? 


ling  per  yard  V 


Ans.  JC3Sfi.  8s. 
OPERATK 

li 


r72£  shillings 


Ans.   .C723. 


OPERATION. 
241 

3 


723  pounds. 


£  386    8s. 

I.     Sold  4S('»  yards  of  calico  at  5  pence  a  yard,  what  did 
it  amount  to?  Ans.  .£10  2s.  (id. 

T>.     Bought  38495  pounds  of  good  middling  cotton  at  7 
pence  a  pound.      .How  much  did  it  CM  - 

Ans.     £1122  15a  :>d. 

U.      What    is    the   value   of  S,">0   ham-Is    ol'  flour  at   34 
shillings  a  harrel  ?  Ans.      £14-1."). 

7.  How   much  will  1S12  tons  of  iron  cost  at  £52,  4s. 
per  ton?  Ans.   £!l4r>S(j  8s. 

8.  Bought  3>  121  pounds  of  cotton  at  9  pence  per  pound 
what  did  it  cost?  Ans.    C1440  15s.  9d. 

74.  MISCELLANKors  PROBLEMS  INVOLVING  THE  I'HIN- 
CIl'LKS  OK  ADDITION,  SUBTRACTION,  MULTIPLICA- 
TION AM)  DIVISION'. 

1.  The  subtrahend  is  210,  and  the  remainder  184,  what 
is  the  minuend  ?  Ans.  400. 

2.  A  irrocer  paid  8350  for  some  tea  and  coffee ;  for  the 
tea  he  paid  $50  more  than  for  the  coffee,  what  did  he  pay 
for  each  ?  Ans.  tea  8200,  coffee  $150. 

3.  John  has  25   cents,  and  James  has  four  times  as 
many  lacking  10  cents,  how  many  cents  has  James? 

Ans.  90  cents. 

4.  A  slate  cost  15  cents;  an   arithmetic  four  times  as 
much  as  the  slate,  and  a  philosoj  hy  twice  as  much,  lacking 
25  cents,  as  the  slate  and  arithmetic.     What  did  they  all 
cost?  Ans.  $2.00. 


Division,  71 

B.     The  sum  of  two  numbers  is  480,  and  their  difference 
is  80,  what  are  the  numbers  ?  Ans  200,  280. 

6.  A  man  purchased  a  horse  and  cow.     For  the  horse 
he  paid  $175,  and  for  the  cow  $110  less  than  for  the  horse, 
what  did  the  cow  cost  ?  Ans.  §65. 

7.  The  less  of  two  numbers  is  224  and  their  difference 
100,  what  is  the  greater?  Ans.  324. 

8.  The  product  of  two  numbers  is  G450,  and  one  of 
the  numbers  is  150,  what  is  the  other?  Aus.  43. 

9.  A  merchant  bought  415  yards  calico  at  10  cts.  per 
yard  and  sold  it  for  13  cts.  per  yard.     How  much  did  he 
gain?  Ans.  812.45. 

10.  The  dividend  is  37500  and  the  quotient  75,  what 
is  the  divisor  ?  Ans.  500. 

11.  A  boy  sold  5  chickens  at  25/  a  piece  ;  8  ducks  at 
50^  each  ;  received  in  payment  3  pigeons  at  30^  each,  and 
the  balance  in  money  ;  how  much  money  did  he  receive  ? 

Ans.  $4.35. 

12.  The  divisor  is  37;  the  quotient  21,  and  the  re- 
mainder 23,  what  is  the  dividend  ?  Ans.  800. 

13.  A  news  boy  sold  20  papers  at  5/  each,  and  with 
the  money  bought  oranges  at  4c  each,  how  many  oranges 
did  he  get?  Ans.  25. 

14.  The  first  battle  of  the  Revolution  was  fought  April 
19,  1775,  how  many  years,  months  and  days  have  passed 
since  then  ? 

15.  H.  Zuberbier  has  an  orange  orchard   consisting  of 
480  trees,  and  each  tree  produces  5  barrels  of  oranges  which 
are  worth  in  the  market  $4  a  barrel ;  what  is  the  value  of 
his  orange  crop  ?  Ans.  81)600. 

16.  C.  Quentell  bought  a  barrel  of  sirop  de  batterie 
containing  43  gallons  at  95/  per  gallon  ;  4  gallons  having 
leaked  out  he  sold  the  remainder  at  $1.U5  a  gallon.     How 
much  did  he  gain  by  the  transactions?    Ans.  $.10  gain. 

17.  W.   C.   Martin   bought  ;354  barrels   of  flour  for 
$2478.     He  sold  the  same  at  $7.50  per  barrel ;  how  mucn 
did  he  gain?  Ans.  $177. 

18.  The  Capital  Stock  of  a  Manufactory  is  $100000 


72         Arithmetical  Exercises  and  Examples. 

which  is  divided  into  200  shares.     What  are  5  shares  worth  ? 

Ans.  £j!f)00. 

19.  J.  Muller  sold  to  O.  Braun  25  barrels  of  apples  at 
$4  per  barrel  and  124  barrels  of  potatoes  at  $3.25  per  bar- 
rel; he  received  in  payment  1  hogshead  of  sugar  containing 
1143  pounds  at  8/  and  the  remainder  in  money  ;  how  much 
money  did  he  receive?  Ans.  $411.5(1. 

20.  A  speculator  bought  528  cords  of  wood  at  $(J.f>0 
per  cord.     He  re-corded  the  wood  so  that  it  measured  57!) 
cords  which  he  sold  at  $6.75   a  cord ;  how   much  did  he 
gain?  Ans.  $476.25. 

CANCELLATION. 

75.  <  ampliation  is  the  proecss  of  shortening  the 
operations  of  division,  or  of  the  indicated  result  of  multi- 
plication and  division  operations  combined,  by  rejecting 
equal  factors  from  both  dividend  and  divisor  or  from  both 
increasing  and  decreasing  numbers. 

The  operation  is  perfoimed  by  drawing  a  line  across  each 
factor  cancelled  or  cut  out. 

7G.    The  Principles  of  Cancellation,  are,  1.  lle- 

jecting  or  Cancelling  a  factor  from  any  number  is  in  effect 
dividing  the  number  by  that  factor.  2.  llejecting  or  can- 
celling equal  factors  from  both  dividend  and  divisor,  or  from 
both  increasing  and  decreasing  numbers  in  an  indicated 
result,  does  not  change  the  quotient  or  result. 

EXAMPLES. 

Divide  7x3x4  by  7>(4 

Operation  by  Cancellation.         Explanation.— In   all    prob- 
y  7  lems  where  we  have  both  niul- 

^  3  triplication  and  division  opera- 

tions  to    perform,    we    use    a 
vertical  or  perpendicular  line 

^  which  we  call  the    statement 

3  Ans.  line.     This  line  is  used  to  facil- 

itate the  work  by  separating  the  dividends  and  divisors,  or  the 
increasing  and  decreasing  numbers.  The  dividends  or  increas- 
ing numbers  are  always  placed  upon  the  right  hand  side  of  the 


Cancellation.  7B 

line  and  the  divisors  or  decreasing   numbers  are  always  placed 
ui)on  the  left  hand  side. 

In  this  example  having  written  the  numbers  that  constitute 
the  dividend  and  divisor,  respectively  upon  the  right  and  left 
hand  side  of  the  statement  line,  we  cut  out  or  cancel  the  equal 
factors  T's  and  4's  in  the  numbers  constituting  the  dividend  and 
divisor  and  thus  obtain  3  the  answer  to  the  problem. 

To   perform   the   work    without   the   aid  of  Cancellation    we 
would  be  obliged  to  make  the  following  figures:  ly^'^='2l,  which 
X4:r=84  the  dividend  ;   then  7X4— 28  the  divisor  ;  then 
28)84(3  Ans. 
84' 


2.      Multiply  25,  48  and  88  together  and  divide  the  pro- 
duct by  the  product  of  10,  30  and  8. 

Operation  by  Cancellation.          AVy/,W/  >/<>//.—  In  this  exam- 
•'  lO'^o     5  P^e  we  wr*te  ^ie  numbers  on 

~         " 


.)  .   ,  <)  the  line  as  above  directed  and 

then  cancel  the  lo,  and  2:.  by 

Kpp  11  f>  ;   then  the  3G  and  48  by  12  ; 

then  the  8  and  88  by  8  ;   then 

110  the  4  and  2  by  2.     This  is  all 

__  that  can  be  cancelled  and  we 

.,  ..,     *  then    multiply    together    the 

50-j    Alls.  5^  .,  .uul  j  l  an  j  jjyide  tne  pr<J_ 

duct  by  3,  and  thus  obtain  the  true  result 

Should  the  student  experience  any  difficulty  in  this  kind  of 
work,  he  should  be  orally  drilled  on  the  factors  of  numbers  and 
composite  numbers*. 


3.     Divide  the  product  of  32  X^  l>y  8 

Operation  by  Cancellation.         E,phni«ti<,n.—  Having  writ- 
-  4  ten  the  numbers  on  the  state- 

-">  merit  line,  we  first  cancel  the 


3     (,) 
|    j^; 


8  and  32  by  8  ;  then  the  9  and 
3  by  3  ;  then  the  1G  and  4  by 


__ 

-)       _    i      A  '*'    ^°W  llav"1S  no  more  num- 

i  -2    Ans-  bers  on  the  increasing  side  of 

the  line  to  cancel  we  multiply  together  the  remaining  numbers 
on  the  decreasing  side  of  the  line  and  thus  produce  the  correct 
result  ,\,. 

In  all  cases  where,  after  cancelling,  no  factor  appears  on  either 


71          Arithmetical  Exercises  and  Examples. 


side  of  the  statement  line,  the  factor  1,  is  always  understood, 
as  being  there.  Its  non-appearance  is  in  consequence  of  not 
having  written  it  when  we  cancelled  a  number  by  itself. 

4.  A  merchant  sold  25  boxes  of  candles  containing  3G 
pounds  each  at  lt>c'  per  pound  and  received  in  payment 
starch  at  (>  cents  per  pound.  How  many  boxes  each  con- 
taining !><)  pounds  did  he  receive?  Ans.  80  boxes. 

Operation  by  Cancellation 

tin; 
p  w  •>»  r, 


80  boxes  Ans. 
Cancel  and  work  the  following  line  statements  or  results: 

12  i:> 


2<; 

.;:» 
ISM 


y  . 


!•  rx; 

784 

in  4 


51 


124 

17 

10 


20 
70 


76 
91 
140 
25 


II  Ans.  70  Ans.          f  Ans.  1921£  Ans. 

5.  Divide  the  product  of  6,  7,  12  and  22  by  the  pro- 
duct of  11,  3,  14  and  8.  ,  Ans.  3. 

().  What  is  the  quotient  of  28x66x7XT8-5-56Xl30 
X42X13?  Ans.  J. 

7.  Multiply  21,  55  and  128  together  and  divide  the 
product  by  14X25X64.  Ans.  6f. 

8.  How  many  bushels  of  corn  at  70^  each  will  pay  for 
140  gallons  molasses  at  (55  cents  a  gallon  ? 

Ans.  130  bushels. 

9.  Bought  420  pounds  of  sugar  at  6  cents  a  pound  and 
gave  in  payment  360  pounds  of  rice.     What  was  the  price 
of  the  riee  ?  Ans.  7  cents. 

10.  Sold  a  drayman  64  bushels  of  oats  at  75  cents  a 
bushel,  for  which  he  is  to  pay  in  drayage  at  50  cents  a  load. 
How  many  loads  must  he  haul  ?  Ans.  96  loads. 

11.  How  many  pounds  of  butter  at  35/  per  pound  will 


Properties  of  Numbers.  75 

pay  for  245  pounds  of  rice  at  5  cents  per  pound  ?  . 

Ans.  35  pounds. 

12.     Paid  65/  for  5  yards  of  calico,  what  will  27  yards 
cost  at  the  same  rate  ? 

Analytic  Solution  by  Can-        Explanation.—  In    all    practical 
cellation.  problems  of  this  kind  we  give  a 

|$j3   13  reason  for  each  step  of  the  opera- 

tion,  and  make  the  whole  state- 
ment  to  indicate  the  final  result 


_ 

^  without    performing    any   of   the 

$0.51  Ans.  intermediate  work.    In  this  prob- 

lem, we  place  the  6-r>?  on  the  increasing  side  of  the  statement 

line  as  our  premise  and  reason    thus:  since  5   yards  cost  <J.V?. 

1  yard  will  cost  i  part  of  it,  and  27  yards  will  cost  27  times  as 

much  as  1  yard. 

13.  If  17  barrels  of  flour  cost  $110.50,  what  will  500 
barrels  cost  at  the  same  rate  ?  Ans.  $3250. 

14.  If  |  of  a  dozen   apples   cost  40  cents,  what  will 
13i-  dozen  cost?  Ans.  $7.20. 

PROPERTIES  OF  NUMBERS. 


DEFINITIONS. 

77.  All  Integer  is  a  whole  number  ;  as  1,  5,  6,  18,  etc. 
Whole  numbers  are  divided  into  two  classes,  Prime  and 

Composite. 

78.  A  Prime  number  is  one  that  can  only  be  divided, 
without  a  remainder,  by  itself  and  1  ;  as  1,  2,  3,  5,  7,  11. 
13.  17,  etc. 

7!>.  A  Composite  number  is  one  that  can  be  divided 
without  a  remainder,  by  some  other  whole  number  than 
itself  and  1  ;  as  4,  0,  12,  15,  24,  etc. 

All  composite  numbers  are  the  product  of  two  jjr  more 
other  numbers. 

Numbers  are  prime  to  each  other  when  they  have  no 
common  factor  that  will  divide  each  without  a  remainder; 
as  0,  13,  20,  etc. 


76         Arithmetical  Exercises  and  Examples. 

SO.      An  Kveil  number  is  one  that  can  be  divided  by  2. 
without  a  remainder;  as  4,  8,  12,  50,  etc. 

81.  All  0(1(1  number  is  one  that  cannot  be  divided  by 
2,  without  a  remainder,  as  1,7,  l'.».   l.">.  l.'J.'J.  etc. 

82.  A   Factor  of  a   Number  is  a  number  that  will 
divide  it  without  a  remainder  or  by  being  taken  an  entire 
number  of  times,  will  produce  it  ;   as  4  is  a  factor  of  1(>,  and 
5  a  factor  of  2.">. 

.Kvery  factor  of  a  number  is  a  divisor  of  it. 

83.  A  Prime  Factor  of  a  number  is  a  prime  number 
that   will   divide  it  without  a  remainder  :   thus,  1.  2.  3  and 
5  are  the  prime  factors  of  .'Jo. 

84.  A  Composite  Factor  of  a  number  is  a  composite 
number   that   will   divide  it  without  a  remainder  ;   thus,  (j 
and  8  are  composite  factors  of    IS 

85.  An  Aliquot  part  of  a  number  is  such  a  part   a> 
will  divide  it  without  a  remainder;  thus,   1,  2,  3,  4,  (\  and 
8  are  aliquot  parts  of  2  I. 

86.  TllO  Reciprocal  of  a  number  is  the  quotient  of  1 
divided  by  the  number  ;  thus  the  reciprocal  of  8  is  1  — 8=|- ; 
and  the  reciprocal  of  \  is  1  -=-  ^  =  4. 

87.  The  Power  of  a  number  is  the  product  obtained 
by  multiplying   the  number  by  itself  a  certain    number  of 
times  ;  thus,  36  is  the  second  power  of  0  ;   125  is  the  third 
power  of  5. 

88.  The  Multiple  of  a  number  is  the  product  obtained 
by  multiplying  the  number  by  any  other  number  any  num- 
ber of  times.     Or  it  is  a  number  divisible  by  a  given  num- 
ber without  a  remainder ;  thus,  14  is  a  multiple   of  7  and 
2.  and  54  is  a  multiple  of  2,  3,  !>,  and  27. 

89.  A  Common  Multiple  of  two  or  more  numbers  is 
a  number  divisible  by  each  of  them   without  a  remainder  ; 
thus,  24  is  a  common  multiple  of  2,  3,  4,  6  and  12. 

00,    The  Least  Common  Multiple  of  two  or  more 


Divisibility  of  Numbers.  77 

numbers  is  the  least  number  that  is  divisible  by  each  of 
them  without  a  remainder;  thus,  12  is  the  least  common 
multiple  of  2,  3,  4,  6  and  12. 

DIVISIBILITY  OF   NUMBERS. 

91.  A  Divisor,  or  measure  of  a  number,  is  any  num- 
ber that  will  divide  it  without  a  remainder  ;  thus,  4  is  a 
divisor  or  measure  of  12,  and  5  is  a  divisor  or  measure 
of  20. 

Cue  number  is  said  to  be  Divisible  by  another  when 
the  remainder  is  0. 

92.  A  Common   Divisor  of  two  or  more  numbers  is 
a  number  that  will  divide  each  of  them  without  a  remain- 
der; thus,  2  is  a  common  divisor  of  12,  18  and  24. 

93.  The  Greatest  Common  Divisor  of  two  or  more 

numbers  is  the  greatest  number  that  will  divide  each  of 
them  without  a  remainder  ;  thus,  6  is  the  greatest  common 
divisor  of  12,  18  and  24. 

94.  Every  number  ending  with  0,  2,  4,  6  or  8,  is  divi- 
sible by  2. 

95.  Every  number  is  divisible  by  4  when  its  units  and 
tens  figures  are  divisible  by  4  ;  thus,    15(3,   264,   34312, 
561308,  are  each  divisible  by  4. 

96.  Every  number  is  divisible  by  8  when   the  units, 
tens  and  hundreds  figures  are  divisible  by  8 ;  thus  3824, 
12512,  190720  are  each  divisible  by  8. 

97.  All  numbers  ending  in  0  or  5  are  divisible  by  5  ; 
thus,  10,  15  and  35  are  divisible  by  5. 

98.  Every  number,  the  sum  of  whose  figures  is  divi- 
sible by  3  or  9  without  a  remainder,  is  divisible  by  3  or  9  ; 
thus,   135,  3456,  12345912,  etc.,  are  each  divisible  by  3 
and  by  9. 

99.  Every  even  number,  the  sum  of  whose  figures  is 
a* 


78          Arithmetical  Exercises  and  Examples. 

divisible  by  3  without  a  remainder,  is  divisible  by  (\ ;  thus 
318,  12414,  etc.,  are  divisible  by  (5. 

Fiucnoxs. 

KM).  A  Fraction  is  one  or  more  of  the  equal  parts  of 
a  unit  of  any  kind,  or  of  a  collection  of  units  taken  tn^-eth'-r. 
Or  more  briefly,  a  part  of  anything,  or  a  numerical  expres- 
sion of  a  part  of  a  unit. 

101.  A  Fractional  Tnit  is  one  of  the  equal  pans 

into  which  any  integral  unit  is  divided.  If  the  i,it«-tirral 
unit  is  divided  into  two  equal  pans,  each  is  called  n  //////'. 
it  into  three,  each  is  called  a  thinl  ;  if  into  four.  ea«-h  is 
called  a  /'nurtli  ;  and  so  on  according  to  the  number  «»f 
parts  into  which  the  integral  unit  is  divided. 

102.  Fractions  are  divided  into  two  kinds.   (1<nnm<ni  or 
Ynlijiir.  and    ]>n-iin<il  fr'nu-tinim. 

Common   Fractions  are  expressed  by  two  numbers,  one 

written  above  the  other,  with  a  hori/.ontal  line  between 
them.  The  number  below  the  line  is  called  the  D>  nnmin- 
<ttur,  and  the  number  above  the  line  is  called  the  .Xmnrrator. 
Thus  2  (one  half,}  J  (three  fourths,)  f  (Jive  .\v>//ix, )  J 
(,«-rrn-  r/V////x,)  and  }  ^  (thirteen  sevciitrrnthx)  are  fractions, 
the  denominators  of  which  are,  respectively,  2,  4,  6,  8,  and 
17.  The  Numerator  and  Denominator  together,  are  called 
the  terms  of  the  fraction. 

The  Denominator  of  a  fraction  shows  the  number  of 
equal  parts  into  which  the  unit  is  divided. 

Thus  in  the  fraction  |  the  8  is  the  denominator  and  shows 
that  the  unit  is  divided  into  8  equal  parts  called  eighths. 

The  Numerator  of  a  fraction  shows  the  number  of  equal 
parts  taken  to  form  the  fraction. 

Thus  in  |-,  the  numerator  is  5  and  shows  that  5  of  the 
8  equal  parts  are  taken  or  expressed  by  the  fraction. 

All  fractions  aritie  from  division  and  are  expressions  of 
unexecuted  division  in  which  the  numerator  is  the  dividend, 
the  denominator  the  divisor,  and  the  fraction  itself  the 


Fractions.  79 

Decimal  Fractions  are  those  in  which  the  denominators 
are  not  generally  expressed,  but  are  always  10,  or  a  power 
of  ten;  thus,  .5,  .75,  .821,  read  respectively  Jive  tenths, 
seventy-five  hundredth*,  and  eight  hundred  and  twenty-one 
thousandths,  are  decimal  fractions.  To  write  these  fractions 
as  common  fractions,  they  would  be  written  thus,  y5^,  T75% 

' 


The  point  (.)  placed  before  the  5,  7  and  8,  in  the  above 
decimally  expressed  fractions,  is  called  the  decimal  point, 
and  is  used  to  abbreviate  the  work. 

103.  CLASSIFICATION  OF  FRACTIONS. 

For  convenience  fractions  are  classed  under  the  following 
heads  :  Proper  Fractions  ;  Improper  Fractions  ;  Simple 
Fractions;  Mixed  Numbers;  Compound  Fractions  ;  Com- 
plex Fractions. 

104.  A  Proper  Fraction  is  one  in  which  the  nume- 
rator is  less  than  the  denominator,  as  i,  f  ,  J. 

105.  An  Improper  Fraction,  is  one  in  which  the 
numerator  is  equal  to  or  greater  than  the  denominator  ;  as 

'••  and  V'. 

10l>.  A  Simple  Fraction  is  one  in  which  both  terms 
are  whole  numbers,  and  may  be  either  a  proper  or  hnpn^n  r 
fraction  ;  as  f  ,  f  ,  fj-  or 

107.  A  Mixed  Number,  is  a  number  composed  of  a 
whole  number  and  a  fraction  ;  as  2],  5:]  and  -IfV 

103.  A  Compound  Fraction  is  a  fractional  part  of 
a  fraction  or  mixed  number  ;  as  -J  of  |  and  2  of  r7^  of  1'J:]. 

109.  A  Complex  Fraction  is  one  that  has  one  or 
more  of  its  terms  fractional  ;  as  :  * 

i         61  3         i          6} 

-   of  —  r  and  -   of  —   of  — 
I         5|  §         II          8 

UO,    The  Reciprocal  of  a  Fractlqn  is  the  result  yf 


^         Arithmetical  Exercises  and  Examples. 

1  divided  by  the  fraction.     Thus  the  reciprocal  of 

-H=*=u. 

111.  The  Value  of  a  Fraction  is  the  result  of  its 
numerator  divided  by  its  denominator.  Thus  J--1. 

2  . 

112.    <;KNKRAL  PRINCIPLES  OF  FRACTIONS. 

1.  Multiplying  the  numerator,  or  di  rid  ing  the  denom- 
inator, multiplies  the  fraction. 

'1.  Dividing  the  numerator,  or  multiplying  the  denmn- 
inator,  divides  the  fraction. 

3.  Multiplying  or  dlrldlny  both  numerator  and  denom- 
inator by  tlie  sumo  number  does  not  change  the  value  of 
the  fraction. 

113     .REDUCTION  OF  FRACTIONS. 

Reduction  Of  Fractions  is  the  process  of  changing 
theiryorw  without  altering  their  c<ilu< . 

114.  A  fraction  is  reduced  to  Higher  Terms   when 
the   numerator  and  denominator   are   expressed  in  larger 
numbers.     Thus,  }=|  or  f  or  T8F  ;  f — £  or  T\  or  -J-J,  etc. 

115.  A  fraction  is  reduced  to   Lower  Terms   when 
the  numerator  and  denominator  are  expressed  in  smaller 
numbers.     Thus  TV— |  or  §  ;  i^— f  or  j. 

111).  A  fraction  is  reduced  to  its  Lowest  Terms  when 
its  numerator  and  denominator  cannot  be  divided  by  any 
number  greater  than  1,  or,  when  its  numerator  and  denom- 
inator are  prime  to  each  other. 

117.  Whole  numbers  maybe  reduced  to  fractions 
having  any  desired  denominator. 

118.    ORAL    EXERCISES. 

1  unit,  abstract,  or  denominate  of  any  kind  equals,  how 
many  halves?  thirds?  fourths?   fifths?  sixths?  sevenths? 
eighths?  ninths? 

2  =how  many  halves?  thirds?  fourths?  fifths?  sixths? 
sevenths?  eighths?  ninths? 


Reduction  of  Fractions. 


81 


3— how  many  halves?  thirds?  fourths?  fifths?  sixths? 
sevenths?  eighths?  ninths? 

4=how  many  halves ?  thirds?  fourths?  fifths?  sixths? 
sevenths?  eighths?  ninths? 

5— how  many  halves ?  thirds?  fourths?  fifths?  sixths? 
sevenths?  eighths?  ninths? 

6— how  many  halves ?  thirds?  fourths?  fifths?  sixths? 
sevenths?  eighths?  ninths? 

119.     What  kind  of  numerical  work  is  the  above  called  ? 

i  —  how  many  fourths?  sixths?  eighths?  tenths? 
twelfths?  fourteenths? 

£  =  how  many  sixths  ?  ninths  ?  twelfths  ?  fifteenths  ? 
eighteenths  ?  twenty-firsts  ? 

J  =  how  many  eighths?  twelfths?  sixteenths?  twen- 
tieths ?  twenty-fourths  ? 

£  =  how  many  tenths?  fifteenths?  twentieths?  twenty- 
fifths?  thirtieths? 

J  —  how  many  sixteenths  ?  twenty-fourths  ?  thirty- 
seconds?  fortieths?  sixty-fourths? 

y1^  =  how  muuy  thirty-seconds  ?  forty-eighths  ?  sixty- 
fourths?  eightieths? 

What  kind  of  numerical  work  is  the  above  called  ? 
Answer  the  following  numerical  questions : 


F' 

120.     What  kind  of  numerical  work  is  the  above  called? 

How  many  halves  =  or  make  a  unit? 

"  u        thirds  = 

"  "      fourths  — 
fifths  = 

"  "  sixths  — 
"  "  sevenths  — 
<'  "  eighths  = 


Arithmetical  Exercises  and  Examples. 


Huw  many  ninths  =  or  make  a  unit? 
"       "       tenths  =     "       "        " 
"       "  elevenths  =     "       "        " 
"     twelfths  =     "       "        " 


121. 


1    unit  equals  how 

many  eighths?     • 

^                  U                  (( 

twelfths?    |'V 

.')  units  equal 

thirds?    ! 

^           a            a 

fourths?  X 

-             «            « 

halves?   i 

1J         " 

halves?    | 

1  '>          «            « 

fourths?     ; 

1  :!               " 

thirds?    4 

I! 

sixths?  ^ 

1  ;|      «        " 

eiuliths?    ] 

;;i           u             c< 

fourths'/    \ 

•>  :           cc             tc 

JO                     /  /                          f  i 

sixteenths?    '/ 

1.1           0        0 

—  ? 


*=? 


r; 

1  B 


122.     What  is  the  value  of  the  following  fractional  ex- 
pressions ? 

if  of  £  of  *=? 


.1  of 


I  of  J=? 
.'>  of  J=? 
i  of  i=? 
ft  of  i=? 

ff=» 

!i  of  J=? 


123.  What  is  the  reciprocal  of  1,  of  2,  of  3,  of  i,  of  :]• 
of  1^? 

What  is  the  value  of  f,  of  \2,  of  *£>  of  V»  of  V  ? 
Analyse  the  fraction  J. 

Analysis: — \  is  a  proper  fraction,  since  the  numerator  is  less 
than  the  denominator;  4  is  the  denominator,  and  shows  that 
the  unit  is  divided  into  4  equal  parts  ;  |  is  the  fractional  un.t, 
since  it  is  ONE  of  the  four  equal  parts  into  which  the  unit  is  di- 
vided ;  H  is  the  numerator  and  shows  that  three  of  these  equal 
parts  are  taken  ;  3  and  4  are  the  terms  of  the  fraction,  and  its 
value  is  less  than  1,  or  unity. 


Reduction  of  Fractions. 


In  like  manner  analyse  the  following  fractions  : 

|5    |j     J         12^    ^5?     |?     4^     |4 

124.     Mentally  add  the  following  fractions  : 


f  +  *  + 


*+»=? 
f+l=? 

i+t=? 

125.     Answer  by  mental  work  the  following  numerical 
questions : 

n  11    9 

»—,*=? 


126.       ORAL  EXERCISES. 

1.  What  will  2}  yards  cost  at  $J  per  yard  ? 

Ans.  SI  ,x. 

Analytic  solution.  —  Since  1  yard  cost  SJ,  J  a  yard  will  cost  J  as 
much,  which  is  $;  ;  and  ]  yards  (2j  yards  reduced  to  halves) 
will  cost  5  times  as  much,  which  is  $1S"  or  $U- 

2.  AYluit  will  2}  pounds  cost  at  7-]/  per  pound  ? 

Analytic  solution.  —  Since  1  pound  cost  -1./'  cents,  J  of  a  pound 
will'  cost  }  part  which  is  y  cents  ;  and  Y  will  cost  11  tiiu^s  's"' 
cents,  which  is  i|j-5-  cents  or  20  J-  cents. 

In  like  manner  solve  the  following  problems. 

3.  At  $3   a  yard   what   will  5J  yards   cost? 

Ans.  t2& 

4.  A  dozen  is  worth  $3o,  what  are  4^  dozen  worth? 

Ans.  $15i-. 

5.  What  is  the  value  of  61   dozen  apples  at  §1  per 
dozen  ?  Ans.  $'2\\. 

(i.     What  cost  82   gross  at  $2J  per  gross? 

Aiis.  S2.-J 


. 

1.     If  ;|  of  a  pound  cost  12  cents,  what  is  1  pound 
worth  ?  Ans.  16  cents. 


S4         Arithmetical  Exercises  and  Examples. 

Analytic  solution. — Since  J  of  a  pound  cost  12  cents,  J  will 
cost  \  of  12  cents,  which  is  4  cents,  and  j  or  1  pound  will  cost 
4  times  4  cents,  which  is  16  cents. 

2.  f  of  a  dozen  cost  $10,   \Uiat  is  the  value   of  2j 
dozen?  Ans.  $371. 

Analytic  solution. — Since  J  of  a  dozen  cost  $10,  J  of  a  dozen 
will  cost  J  as  much,  which  is  $5,  and  jj  or  a  whole  dozen  will 
costs  times  as  much,  which  is  $15;  and  since  1  dozen  cost 
$1:"),  j  dozen  will  cost  £  as  much,  which  is  $7J  and  }  dozen  will 
cost  5  times  as  much  which  is  $37J. 

In  like  manner  solve  the  following  problems : 

3.  If  |  of  a  yard   cost  $2],  what  will  1  yard  cost  ? 

Ans.  $0. 

4.  If  |  of  a  yard  cost  $2£,  what  will  \  of  a  yard  c 

Ans.  8} 

5.  4  of  a  number  is  15,  what  is  the  number? 

Ans.  20. 

6.  If  s  of  a  number  is  8,  what  is  13  times  the  number  ? 

Ans.  LM. 

7.  If  %  of  a  dozen  cost  $8,  what  will  £  of  a  dozen  cost 
at  the  same  rate  ?  Ans.  $9. 

8.  What  part  of  4  is  3  ?  Ans.  J. 

Analytic  solution. — Here  by  the  terms  of  the  question  we  have 
3  to  divide  or  measure  by  4,  and  by  the  exercise  of  our  reason 
we  proceed  thus  :  since  3  is  equal  to  1,  3  times,  it  is  equal  to  4 
J  of  3  times,  which  is  J.  Or  thus.  Since  1  is  J  of  4,  3  is  3 
times  J,  which  is  J. 

9.  What  part  of  5  is  5  ?  Ans.  T^. 

Analytic  solution. — Since  $  is  equal  to  1,  ij  of  a  time,  it  is  equal 
to  5  the  i  part  of  f  of  a  time,  which  is  f$. 

10.  What  part  of  |  is  7  ?  Ans.  8}. 

Analytic  solution. — Since  7  is  equal  to  one  7  times,  it  is  equal 
to  1,  5  times  7  which  is  35,  and  to  4  instead  af  i  to  \  part  of  35, 
which  is  8J. 

What  part  of  f  is  f  ?  Ans.  1 1  f 

Analytic  solution. — Since  |  is  equal  to  1,  $  of  a  time,  it  is 
equal  to  J  8  times  f  which  is  4g°,  and  to  f  instead  of  J,  to  J  part 
of  \°,  which  is  | J  or  IJf. 


Reduction  of  Fractions.  85 

12.  What  part  of  $  is  4?  Ans.  ff 

13.  What  part  of  3  J  is  2  H  Ans.  T9¥. 

14.  What  part  of  5  is  J  of  2  ?  Ans.  TV 

15.  What  part  of  4  is  f  of  f  ?  Ans.  ^%. 

16.  9  is  i  of  what  number?  Ans.  72. 

Analytic  solution. — Since  9  is  ^  of  a  number,  |  or  the  whole 
number  is  8  times  9  or  72. 

17.  13  is  \  of  what  number?  Ans.  91. 

18.  21T\  is  |  of  what  number  ?  Ans.   106J. 

19.  ^  is  -}-  of  what  number?  Ans.   f. 

20.  24  is  £  of  how  many  times  3  ?  Ans.   10. 
Analytic  solution. — Since  24  is  i  of  the  number,  ^  is  J  part  of 

24  which  is  6,  and  |-  or  the  whole  number  is  5  times  6,  which  is 
30  ;  and  as  30  is  equal  to  3,  10  times,  therefore  '?4  is  $  of  10 
times  3. 

21.  32  is  \  of  how  many  times  8  ?  Ans.  7. 

22.  28  is  T7^  of  how  many  times  12  ?  Ans.  5. 

23.  \  of  48  is  %  of  what  number  ?  Ans.  54. 

Analytic  solution. — Since  48  is  the  whole  of  a  number,  J  of 
the  number  is  \  part  of  48,  which  is  1 2,  and  f  is  3  times  12,  which 
is  30 ;  and  since  36  is  jj  of  an  unknown  number,  J  of  it  is  J  of 
.'56,  which  is  18,  and  |  or  the  whole  number  is  3  times  18,  which 

is  54. 

24.  |  of  63  is  T4r  of  what  number?  Ans.   154. 

25.  8  of  %  of  64  is^of  what  number?  Ans.   104. 

26.  i  of  ^  of  42  is  J-  of  what  number  ?       Ans.  7£. 

27.  J  of  32  is  f  of  4  times  what  number?    Ans.  9. 

Analytic  solution. — Since  32  is  the  whole  of  a  number,  \  of  the 
number  is  \  part  of  32,  which  is  8,  and  J  is  3  times  8,  which  is 
24  ;  and  since  24  is  §  of  4  times  an  unknown  number,  j  of  4 
times  the  number  is  J  of  24,  which  is  12,  and  f  or  the  whole  of 
I  limes  the  number  is  3  times  12,  which  is  36  ;  and  since  36  is 
4  times  the  number,  \  of  3£,  which  is  9,  is  the  required  number. 

2S.  I  uf  40  is  f  of  7  times  what  number?  Ans.  6. 
25).  i  df .")()  is  J  ot  6  times  what  number?  Ans.  12. 
30.  f|  of  {  of  t>6  is  3|  of  3  times  what  number? 

Aus.  3. 


86          Arithmetical  Exercises  and  Examples. 

31.     What  is  the  i  and  \  of  a  J,  of  §  of  15  ? 

Ans.  5. 

127.  GREATEST  COMMON  DIVISOR. 

For  a  definition  of  a  divisor,  a  common  divisor,  and  the 
greatest  common  divisor,  see  page  77. 

1.  What  is  the  greatest  common  divisor  of  42,  56  and 
210? 

OPERATION.  Rjcp!(in<itwn.—ln  all  problems  of 

42.  56.  210  tn'8   kind    we   first   divide  by  any 

'        factor  that  will  divide  all  the  num- 

91    9Q    1  n^  bers  ;  then  we  divide  in  like  man- 

6l.  £o.  1UD  ner  tQe  successive  quotients  thus 

obtained,  until  we  obtain  quotient? 
3      4      15  that  have  no  common  factor;   then 

we   multiply  all   the    divisors   to- 

j  *     i     ^  gether  and  in  the  product  we  have 

the  greatest  common  divisor. 
When  there  is  no  number  greater  than  1  that  will  divide  all 
the  numbers  without  a  remainder,  then  1  is  the  greatest  com- 
mon divisor. 

When  there  are  two  large  numbers  the  operation  may  be 
more  easily  performed  by  first  dividing  the  larger  number  by 
the  smaller,  and  if  there  is  a  remainder  divide  the  preceding 
divisor  by  it,. .and  thus  continue  until  there  is  no  remainder. 
When  there  are  more  than  two  numbers,  proceed  as  with  two, 
and  then  with  the  greatest  common  divisor  of  the  two  and  one 
of  the  other  numbers,  and  thus  continue  until  through  all  the 
numbers.  The  last  divisor  will  be  the  greatest  common  divisor. 


Greatest  Common  Divisor. 


87 


2.     What  is  the  greatest 
common  divisor  of  88  and  24  ? 
Ans.  8. 

OPERATION. 

24)88(3 
72 

16)24(1 
16 

8)16(2 
16 


3.  What  is  the  greatest 
common  divisor  of  195,  285, 
and  315?  Ans.  15. 

OPERATION. 

285)315(1 
285 

30)285(9 
'270 

15)30(2 
30 

15)195(13 
15 

45 
45 


What  is  the  greatest  common  divisor  of  the  following 
numbers  ? 

4.  Of  441  and  567  ?  Ans.  63. 

5.  Of  90,  315  and  810?  Ans.  45. 

6.  Of  654,  216,  and  108?  Ans.  6. 

7.  Walker  has  25  and  Caruthers  45  dimes,  how  shall 
they  arrange  them  in  packages,  so  that  each  shall  have  the 
same  number  in  each  package  ?   Ans.  5  in  each  package. 

8.  A  planter  has  697  bushels  of  corn  and  204  bushels 
of  rough  rice,  which  he  wishes  to  put  in  the  least  number 
of  bins  containing  the  same  jnumber  of  bushels,   without 
mixing  the  two  kinds.     Kow  many  bushels  must  each  bin 
hold?  Ans.   17  uushels. 

9.  A  Commission  Merchant  has  2490  bushels  of  wheat, 
1886  bushels  of  corn  and  8438  bushels  of  oats,  which  he 
wishes   to  ship  in  the  least   number  <»f  sacks  of  equal  size 
that  will   exactly   hold  either  kind   of  grain.     How  many 
sacks  will  he  require  ?  Ans.   6407. 


Arithmetical  Exercises  and  Examples. 

128.  LEAST  COMMON   MULTIPLE. 


For  a  definition  of  a  Mufti  l<\  <i  Cnmmun  Miilriplf.,  and 
a  Lettsf  CotniiHtn  Mn/fi^ir.  see  pajje  76. 

1.  What  is  the  least  ennimon  multiple  <»:  .").  6,  ^.  21.  2* 

"I'KKATION.  fi^flllfltfon.—  In  all   prob- 

-     5.  f>.  S.  21  lemg  Of  thjs    kjn<i   we  fir!;t 

arrange   the    numbers   on  a 

'2)  5   3   4    -1     14  horizontal  line,  and  then  di- 

___  vide    by    the    */nnl/<'itt    prime 

.,     ,         >;    %>        _  number  that  will  divide  two 

1    f>    -    -'       *  or  more  without  a  remainder 

and  write  the  quotient  and 

7  i  .">    1    '2      7     7  undivided  numbers  in  a  line 

___  below  ;  this  process  of  divid- 

-    j    .;,      i       •  ing  we  continue  until   there 

are  no  two  numbers  that  can 
be  divided  by  the  same  num- 

2X2X3X7X^X2=840  ADS.    her    without   a    remainder; 

then  we  multiply  the  divi- 

sors and  the  numbers  in  the  last  line  together,  and  the  product 
is  the  least  common  multiple. 

When  there  is  any  number  that  will  divide  any  of  the  others 
without  a  remainder  it  may  be  cancelled  before  commencing  to 
divide. 

2.  What  is  the  least  common  multiple  of  4,  9,  12,  15, 
and  24  ?  Ans.    -JGO. 

What  is  the  least  common  multiple  of  the  following: 

3.  Of  8,  4.  Hand  HO?  Aus.   360. 

4.  Of  50.  27,  3,  45  and  63  ?         Ans.  9450. 
:>.  Of  21,  36,  11  and  22?          Ans.  2772. 
(>.  Of  800,  600,  10,  40  and  12?      Ans.  2400. 

7.  Of  8,  18,  20  and  70  ?  Ans-  2520. 

8.  A  drayman   has  2  drays  and  2  floats  ;  on  1  dray  he- 
can  haul  9  barrels  of  flour,   and  on   the  oth  T  12  barrels: 
on  1  float  he  can  haul  18  barrels,  and  on  the  other  21  bar- 
rels ;   what  is  the  least  number  of  barrels   that   will  make 
full  loads  for  either  of  th  ;  drays  or  floats. 

Ans.  252. 


Reduction  of  Fractions.  89 

9.  A  fruit  dealer  desires  to  invest  an  equal  amount  of 
money  in  oranges,  peaches  and  grapes,  and  to  expend  as 
small  a  sum  as  possible  ;  the  price  of  oranges  is  $2.40  per 
box  ;  peaches  $1.60,  and  grapes  for  a  medium  article,  90 
^.,  and  for  first  quality,  $1.20  ;  of  these  two  qualities  the 
fruit  dealer  took  the  cheaper.  How  much  more  money  did 
he  invest  than  he  would  had  he  taken  the  grapes  at  $1.20 
per  box  ?  Ans.  $28.80. 

WRITTEN    EXERCISES. 
REDUCTION  OF  FRACTIONS. 
129.      To  rc(ln<c  fraction*  to  tlidr  lowest  terms. 
Reduce  |^  to  its  lowest  terms. 

FIRST    OPERATION.  Explanation.— In  all  problems 

2)||  =  !~| ;   4)|-|  =  I  Ans.   of  this  kind  we  divide  both  the 

numeratorand  denominatorsuc- 

SECOND    OPEATION.  cessively   by   each   of  the   com- 

Q\56         7    A  mon   'actors  they  contain.     Or 

'64  ^~  i"  Ans-  as  shown  in  the  second  opera- 

tion we  may  produce  the  same  result  with  less  figures,  by 
dividing  both  terms  of  the  fraction  by  their  greatest  common 
divisor. 

By  this  reduction  we  change  the  form  of  the  fraction  |-J, 
but  we  do  not  alter  or  change  its  value,  lor  the  fractional 
unit  of  the  resulting  fraction  (i)  is  8  times  as  great  while 
the  number  taken  is  ^  as  great. 

When  the  t  rms  of  the  fraction  have  no  common  factor 
greater  than  1,  the  fraction  is  in  its  lowest  terms  and  is 
called  an  irr«/nci/>/c  fraction. 

The  object  of  reducing  fractions  to  their  lowest  terms  is  to 
enable  us  to  more  easily  and  readily  understand  their  value. 

Reduce  the  following  fractions  to  their  lowest  terms : 
-•     if- ii-ff-ff  Ans.   J,f,  J.ffr. 

a-    H*.  Hi  itt-  An«-  A,  t, 


90         Arithmetical  Exercises  and  Examples. 


«• 


130.      To  reduce  whole  or  inir,<l  nnml>*>r$  to  improper 
fractions. 

1.  Reduce  5ii  to  an  improper  fraction  or  to  tkml*. 

OPERATION.  Explanation. — In  all  problems  of  this 

5jj  kind  we  reason  thus:  Since  there  are 

3  thirds  in  every  unit  or  whole  num- 
ber, in  5  units  there  are  5    times  as 
•JT   Ans.          many,  which  is  15  -(-  the  $  make  ls7-. 

2.  Reduce  9  to  a  fraction  whose  denominator  is  6. 

OPERATION. 

3=z:-5/-  Ans. 


1U.      Reduce  the  following  numerical  expressions  to  Im- 
proper fractions. 


3. 

8J 

Ans. 

•¥"• 

8 

71} 

Ans. 

J-f3- 

4. 

161 

Ans. 

¥• 

9. 

68$ 

Ans. 

479 

5. 

17} 

Ans. 

£• 

10. 

2183? 

Ans. 

8.J3_5 

6. 

32| 

An-.  J 

-fi. 

11. 

^i 

Ans. 

w- 

7. 

4354 

Ans.  & 

12. 

Ans.  1 

13. 

Reduce 

14  to  a  fraction  whose  denominator  is  9. 

14. 

u 

37         " 

u 

u             u 

a 

24. 

15. 

11 

54}       " 

u 

"        u 

u 

10. 

132.      To  redu  e  improper  fractions  to  whole  or  mixed 
numbers. 

Reduce  -^  to  a  mixed  number. 

OPERATION.  Explanation. — In  all  problems  of  this 

17 .—.^.i    Ans          kind  we  reason  thus.     Since  there  are 

*  4  fourths  in  1  unit  or  whole  number,  in 

°  11  fourths  there  areas  many  units  as  17 

17-i-4— -4ff  Ans.      js  equai  to  4,   which  is  4  times  with    1 

remainder,  or  altogether  4}  as  the  proper  quotient  or  answer. 


Reduction  of  Fractions.  91 

Reduce  the   following   improper  fractions   to   whole   or 
mixed  numbers. 

Ans.  4.          6.      %9-  Ans.  3f. 

Ans.  5|.        7.      f  g  Ans.  5TV 

Ans.  48.          8.     ^8T3-  Ans.   9 
Ans.  18}.        9.  *WP-      Ans. 


133.  To  reduce  Compound  Fractions  to  Simph'  Frac- 
tions. 

Reduce  \  of  f  of  J  to  a  simple  fraction. 

OPERATION  Explanation.  —  In  all  problems  of 

i  v  3  v     _  21     A    <  this  kind  we  multiply  together  all 

X-j-Xa      -g-Q   Ans.         the  nunierators  for  a  new  nume- 
rator and  all  the  denominator  for  a  new  denominator. 

When  a  compound  fraction  contains  whole  or  mixed  numbers 
they  must  first  be  reduced  to  improper  fractions. 

When  there  are  common  factors  in  both  terms  of  a  compound 
fraction  they  should  be  cancelled  before  multiplying.  By  this 
cancelling  the  common  factors  the  work  is  shortened,  and  the 
result  unchanged  for  the  reason  that  dividing  both  terms,  of  a 
fraction  by  the  same  number  does  not  alter  its  value. 

2.  Reduce  §  of  |  of  f  to  a  simple  fraction. 

OPERATION. 

$     ?S     5      5 
-X-X-=—  Ans. 
£     4     8     16 
2 

3.  Reduce  |  of  71  of  |  of  4  of  ^  to  a  simple  frac- 
tion. 

OPERATION. 


-X—  X-X-X—  =-    Ans. 

0    9    p  .  i  .  #  a 

9         3 

4.     Reduc'j  the  following  compound  iractions  to  simple 
ones. 


Arithmetical  Exercises  and  Examples. 

5.  £  of  j|  of  f^.  Ans. 

6.  i  of  §  of  T^-.  Ans. 

7.  |  of  3J  of  J.  Ans.   1 

8.  i  of  Ans.  2^, 
i».  &  »*'  '•"''•  Ans.  54 

in.  ii  of  ff  of  17 J.  Ans.  6. 

11.  iXiXfV-  Ans. 

12.  5X^XA-  Ans. 


KM.        To  red  nee  fraction*  of  different    denominator*  to 

equivalent  fraction*  of  a  common  denominator  or  of  ?/!<• 

least  cummnn  denominator. 

1  >">      Definitions  and  Principles,  pertaining  to  this 

kind  of  reduction  of  fra  -lions. 

KM.     A  Common  Denominator  is  a  denominator 

common  to  two  or  more  fractions. 

l  >T.     The    Least  Common   Denominator  of  two 

or  more  fractions  is  the  least  denominator  to  which  all   the 
fractions  can  be  reduced. 

i:K  A  Common  Denominator  of  two  or  more 
fractions  is  a  common  multiple  of  their  denominators-,  and 
the  Least  Counmm  /)<  nominator  of  two  or  more  fractions 
is  the  least  common  multiple  of  their  denominators,  for  the 
reason  that  all  higher  terms  of  a  fraction  are  multiples  of 
its  corresponding  lower  or  lowest  terms. 

1.  Reduce  J,  J,  and -J,  to  equivalent  fractions  having 
a  common  denominator. 

OPERATION.  Explanation.-^^} 

35  it  -$1  problems  of  this  kind 

3X4X8=      96  common  denominator  we  obtain  the  common 

$  of  )       =32  hence  fjL  equivalent  of  J  denominator  by  mul- 

J  of    -91  -_-^72  hence  «,  equivalent  off-  "plying  together  the 

,,   i                                           i   .                    *  denominators  of    all 

-I  of    )         =84  hence  |f,  equivalent  of  $  lhe  fractions     Then 

to  find   the   respective   numerators  we  take  such  a  part   of  the 


Reduction  of  Fractions.  93 

common  denominator  as  the  respective  fractic  ns  are  parts  of  a 
unit,  as  shown  in  the  operation. 

Reduce  the  following  fractions  to  equivalent  fractions, 
having  a  common  denominator. 

2.  "f  |,  and  f  Ans.  r<&,  tffr,  and 

3.  A  *,  and  tV  Ans.  Ml,  «*,  and 

4.  ,«,,  and  ff  Ans.  ||f ,  and  £ff 

5.  J,  A,  A.  H.  and  f 

Ans.  iHH.  ?W&>  MM!,  ftti*  and 

6.  T^,  i,  5,  and  34. 

Ans.  T%<V,  TVA,  fWV  and 

7.  Reduce  i,  :1  and  f  to  equivalent  fractions  having  the 
least  common  denominator. 

OPERATION. 

2  :-!.  4.  8 


L;  8,  2.  4 


3    1    2 

X2=-4  Least  Common  Denominator. 
i  of   ^         =8  hence  ^  is  the  equivalent  of  J. 
J  of    >  24=18  hence  |~f  is  the  equivalent  ol    ,. 
A  of    J       -~21   hi'iKT  r!  j  is  the  equivalent  of  i. 
Explanation. — In  all   problems  of  this  kind  we  first   find  the 
least   Common   Mn/t>ji/c  of  the  denominators  of  all  the  fractions 
as   explained    in   article    128  page    88   which  is   the    h.ast  com- 
mon  denominator.     Then,   having   the    least    common    denomin- 
ator to  find   the  respective  numerators  we  take  such  a  part  of 
tht'  least   common   denominator   as  the  respective  fractions  are 
parts  of  a  unit,  as  shown  in  the  operation. 

Reduce  the  following  fractions  to   equivalent   fractions 

having  a  least  common  denominator. 
8.     *,  |and  f 
D.      ^,.   1  and  /',.. 

10.  ^.  Uand 

11.  A-i;^  H«"«i* 

12.  :,  . 

18.      Jf 


94          Arithmetical  Exercises  and  Examples. 

14.  a.],  :(,  Ji  and  ;>.  Ans.  ff,  ^  -^  and  if 

15.  T8r°r,  3,  \  and  ||. 

A'ns.    If™,    5328,  yWff  and  1665. 

DENOMINATE  FRACTIONS. 

I3t».    A  Denominate  Fraction  is  one  whose  unit  is 
denominate.     Thus  J  of  a  yard  is  a  denominate  fraction. 

140.      To  reduce  "  </<  u<>niin<it<>  fr<i<-t>nn  from  a  greater 
unit  tn  n  A.s-y. 

1.     Reduce  j  «»{   a  yard  to  inches. 

OPERATION.  I-.r^annUnn.—lu  all  problems  of 

this  kind  we  multiply  the  fraction 
3  by  the  units  of  the  scale  to  which 

•;  ;;  it  belongs  until  we  reach   the  unit 

^ required. 

In  this  example  we  multiply  the 


2<   inches.    Ans.    ;;.  v;ir,j  hy  •>   to  re(iuce  it  to  feet, 
then  by  12  to  reduce  it  to  inches,  the  unit  required. 

2.      In  I  of  a  ton  h«»w  many  ounces?       Ans.   2SIMK). 

141.      To  reduce  (t  denominate  fraction  from  a  /r.sx  unit 
to  a  yrrutt  r. 

1.     Reduce  I  of  a  pound  to  a  fraction  of  a  ton. 

OPERATION.  Kxpl<in,ition.—l*    all    problems    of 

»  p  this  kind  we  divide  the  fraction    by 


25 

•_>    4 


the  units  of  the  scale  to  which  it  be- 
lon^s  until  we  reach  the  unit  requir- 
ed.  Tn  this  example  we  divide  the  f 
pound  by  25  to  reduce  it  to  quarters  ; 
then  by  4  to  reduce  it  to  hundred- 


TWfr  ton  ^ns-  weights  ;  and  then  by  20  to  reduce  it 
to  tons,  the  unit  required. 

2.  Reduce  £  of  a  penny  to  a  fraction  of  a  pound  ster- 
ling, Ans.  jrfa. 

ADDITION  OF  FRACTIONS. 

142.  Addition  of  fractions  is  the  process  of  adding  two 
or  more  fractional  numbers  of  the  same  kind,  or  of  the 
same  denomination. 


Addition  of  Fractions. 


95 


(1.)     Add  Jf.f  |f  and  J  together. 
OPERATION. 

;2   3  4-  &  7 


p  p  p 


Whole  numbers 
1 
1 
1 
1 


Explanation.  In 
this  example  there 
are  no  two  frac- 
tions alike,  hence 
they  cannot  be  ad- 
ded until  we  shall 

Fractions  ob-    have    reduced     them     to 
tained  in  addino-fractions  of  the  same  kind. 
°To  facilitate  and  simplify 
g  the  operation,  we  here  re- 

ji  fi  p  p  JLJ      duce    and    add    but    two 

— fractions  at   a    time,   and 

we  first  select  such  two  as 
may  be  the  most  easily 
reduced  and  added.  Ac- 
cordingly, by  inspection, 
we  select  the  J  and  }  as  the  fractions  to  first  reduce  and  add; 
and  by  the  exercise  of  our  reason  we  see  that  \  is  equal  to  f 
which,  added  to  the  J,  make  J,  which,  for  the  reason  that  | 
make  1  is  equal  to  1  and  J.  We  set  the  1  in  the  column  of 
whole  numbers,  and  the  J  in  the  column  of  fractions.  We  then 
cancel  the  £  and  J,  and  select  the  3  and  J  as  the  next  two  frac- 
tions to  reduce  and  add.  Again,  we  see,  reasoning  analytically, 
that  J  is  equal  to  £  and  f  are  equal  to  J,  which,  added  to  the  f, 
make  |,  which  is  equal  to  1  and  £,  which,  reduced,  equals  J. 
The  1  we  set  in  the  column  of  whole  numbers,  and  the  J  in  the 
column  of  fractions,  and  cancel  the  3  and  ;].  We  next  add  the 
\  and  £,  and  by  our  reason  we  see  that  J  is  equal  to  |,  which, 
added  to  the  £,  make  J,  which  we  set  in  the  column  of  fractions, 
and  then  cancel  the  J  arid  £.  We  then  select  the  J  and  |  as  the 
next  two  fractions  to  add,  and  reducing  tha  J  to  8ths,  we  see 
by  our  reason  that  £  is  equal  to  f  and  }  are  equal  to  3  times  as 
many,  which  is  f,  which  added  to  the  |  make  ^3  which  is  equal 
to  1  and  §;  we  set  the  1  in  the  column  of  whole  numbers  and 
the  f  in  the  column  of  fractions,  and  cancel  the  J  and  |  We 
then  proceed  to  reduce  and  add  the  t\vo  remaining  fractions  $ 
and  £.  I>y  inspection,  the  exercise  of  our  reasoning  faculties, 
and  the  use  ot  our  knowledge  of  the  principles  of  numbers  as 
contained  in  the  preceding  \vork,  we  see  that  the  i  and  f  are 
not  only  not  alike, but  that  we  can  neither  reduce  the  1  to  8ths  nor 
the  £  to  5ths,  and,  therefore,  before  we  can  add  them  we  must 
reduce  both  the  •  and  •§  to  equivalent  fractions  of  the  same 
kind,  or  whose  denominators  are  alike.  To  do  this,  we  first 
observe  that  the  denominators  are  not  divisible  by  the  same 
number,  greater  than  1,  and  hence  the  product  of  ;-'^rn.  4fl 


96  Arithmetical  Exercises  and  Examples. 


is  the  least  number  that  both  of  the  fractions  are  reducible  to, 
or,  in  other  words  their  product  40  is  the  least  common  deno- 
minator of  the  two  fractions.  Having  this,  we  next  reduce  the 
$  and  |  to  40ths,  and  by  our  reason  we  see  that  -J-  is  equal  to 
480  and  |  are  equal  to  4  times  as  many,  which  is  {§  ;  then  that 
|  is  equal  to  fa  and  £  are  equal  to  ">  times  as  many,  which  is 

!fr  w  ich  added  to  the  Jj-  make  J£,  which  for  the  reason  that 
$  make  a  whole  one,  is  equal  to  I  and  .},\.  which  we  place  in 
their  respective  columns  and  cancel  tin-  i  and  J.  The  opera- 
tion c1'  adding  the  fractions  is  now  completed,  and  by  adding 
the  whole  numbers  and  annexing  the  remaining  fractions,  we 
have  as  the  correct  result 

Tht-  foregoing  problem  illustrates  the  most  rational,  easy  and 
rapid  system  of  adding  fractions  known,  and  as  fractions  are 
so  indispensable  and  of  so  frequent  occurrence  in  practical  life, 
the  principles  involved  in  the  system  should  be  thoroughly 
understood. 

In  practical  work,  we  would  very  much  shorten  the  opera- 
tion by  adding  several  fractions  at  once,  and  mentally  perform- 
ing the  most,  if  not  all  of  the  reduction  and  addition  work, 
without  stating  the  results.  Thus,  in  the  above  problem,  we 
would  add  the  J,  J  and  J  at  once.  We  can  instantly  sec-  that 
their  sum  is  y  or  2J,  and  without  naming  or  setting  the  2J,  we 
add  to  it  mentally  the  result  of  3  and  j,  whi'.-h  we  mentally  see 
is  |  or  1J,  making  3f,  which  are  the  only  figures  we  set.  Thus 
all  tl.e  fractions,  except  •£  and  £,  are  added  at  one  mental 
operation.  Then  we  mentally  add  the  sum  of -J  and  f  by  the 
same  process  of  reasoning  as  given  in  the  illustration  of  the 
above  example,  and  obtain  the  correct  result  4JJ. 


(2.)     Add  |,  %  and  ^  together. 

OPERATION. 

Fractions  obtained 
by  adding. 


Wholn 
numbers. 


4 

27 


9  ;; 

28  288 


209 


Explanation.  By  in- 
spection and  reason  we 
see  that  there  are  no 
two  fractions  alike,  and 
that  we  cannot  reduce 
either  of  them  to  an 
equivalent  denomina- 
tion of  any  other;  there- 
fore we  select  the  small 
est  two,  f  and  |,  and 
reduce  them  to  equiva- 
lent fractions  of  the 
same  kind,  or  of  the 

same  denominator,  which  wo  find,  by  multiplying  the  denom- 
inators together,  to  be  :•*•;. 


497 
„  0  ijf 


Ans. 


Addition  of  Fractions. 


97 


We  now  find  by  reasoning  as  in  the  preceding  example,  that 
J  and  J  are  equal  respectively  to  f  J,  and  ff ,  and  collectively  to 
f|,  which  is  equal  to  1  and  if.  Then  proceeding  as  in  the 
first  case,  we  add  the  T8T  and  j|. 

(3.)     Add  J  of  f  of  2i,  1  f  and  ^  together. 


OPERATION. 

Statement  >h>wing  the  reduction 
of  the  fractions. 

3 
1     £     ^     3     5     9 

-x-x-  -  

%    $    4     8     6   23 

Statement  showing  the  result  of  the. 
reduction  and  tl\e  iitMt'tioii  of 
fractions. 

Whole  I'nu-lioMs  ol.tjiim-d 

numbers,  by  ml <liiii;. 

t  9     ;  -/:> 


20  L' 


8 


Explanation.  Here  we 
have  compound  fractions 
and  mixed  numbers,  and 
before  adding,  we  reduce 
the  mixed  numbers  to  im- 
proper fractions,  and  the 
compound  fractions  to 
simple  ones.  Then  we  add 
the  J  and  f,  which  are 
equal  to  1  and  J  ;  then 
the  |  and  J,  which  are 
equal  to  || ;  then  the  f  J 
and  ^3-,  which  are  equal 
to  1  and  ^I'jj.  Then  adding 
the  whole  numbers,  and 
annexing  the  fractions,  we 
have  2JJi|  as  the  correct 
result. 


EXAMPLES. 

(4.)  Add  },  },  f,  f  I,  and  ^. 

(5.)  Add  ij  and  Jf 

(6.)  Add  J,  f,  I,  and  ^. 

(7.)  Add  f ,  |J,  ||  and  |f 

(8.)  Add  f ,  ,%  H  and  fj- 

~(9~)  f  of  iJ  and  21  of  4  of  ^ 

(10.)  Add  2J,  6|,  5|  and  21. 


Ans. 
Ans.  2 
Ans.  2 
Ans.  Sj 
Ans. 

jVns.    f| 
Ans.    17 


s.        Arithmetical  Exercises  and  Examples. 

(1  i  and  .jV. 

nd  f. 

,/„.(;£,  IS^and  2^. 


i .-). 

and  in;. 

and  -VJ. 

i  :>." 
,327^  and25i, 

lulu  of  in  sacks 

wreigl)  ;:.[   .  1  i  1571,  L52J,  1  Uf, 

and  Kil  ,;  pounds?     Ans,   L53SH  ll.s. 

Of  .',  and  2j  Of    -\  Of  1.     Ans.    H. 
Add  3-  of  ^  of 'l  and  n  nf  A  of  £.    Ans.    2||. 
IIo\v  many  vards  in  s  l»,»lts  of  dmncstic.  nieasur- 
tfi    follows:    101,39),    i:  ii!' ,!;.    and 

yards?  Ans.  328|f. 

I'D.       14 feags  of  Coffee  weigh  as  follows:   Kii^7,,.  i 

.  164J,  1654,  164|,  165J,  L62J,  \M&.  164f,  L65|, 
Hi.")  i.  1  (i  1 }  /.  and  1  ('».") -I   pounds:   how  many  pounds  in  all? 

Ans.   ^:-iOl. 

V    merchant  nought    ll."):)',    pounds  of  rice  for 
8i)-l;   STl'J    pounds  of  su-ar  for    s^7^:    ."isoji    pounds  ol' 
roifee  for  -Sii");;  'Ji-[(}\    pounds  of  cheese   for  ST5-];  and 
]»ounds   :  f  uraham   flour  for   8 IS:.      What  was   the 
total   number  of  pounds,   and  the   total   cost  of  all  he  pur- 
chased? Ans.  3254 f  pounds,  $357-$- cost, 
,,,and  :-]'J  of  ^  of  1J. 

Ans. 
.     Add  ;v,  14-  and  }  of  3.    Ans. 


Subtraction  of  Fractions.  99 

SUBTRACTION  OF  FRACTIONS. 

114.  Subtraction  of  fractions  is  the  operation  in  num- 
bers of  finding  the  difference  between  two  fractional  num- 
bers that  are  of  the  same  denomination ;  it  is  hence  the 
converse  of  addition. 

(1.)  What  is  the  difference  between  J  and  f?  Ans.  ^ 
OPERATION  Explanation  Here  we  see 


f  =  15 


Ans. 


or    f         4  that  the  denominations  are 

^5       IQ  not  the  same,  and  there- 

fore,  before  we  can  sub- 
tract, we  must  reduce  the 
•£-0-  Ans  fractions  to  a  ccmmon  de- 


nominator.   By  inspection 

and  in  accordance  with  the  principles  as  explained  in  the  first 
problem  of  addition,  we  see  that  the  least  common  denominator 
is  20 ;  then,  that  J  are  equal  to  ^  and  that  |  are  equal  to  if, 
and  that  the  difference  is  2^. 

(2.)     From  28 1-  take  71.  Ans.  21 1. 

OPERATION.  Explanation  In  perfonn- 

28$  ing:  the  operation  of  the 

71  question  before  us,  we  first 

observe  that  the  fractions 

which  constitute  a  part  of 

21i  Ans.  the    numbers  to   be  sub- 

stractv-d    are    not   of    the 

Bftme  kind  or  denomination,  and  hence,  before  we  can  peifrrm 
the  work,  we  must  reduce  them  to  equivalent  fractions.  \Ve 
next  observe  that  the  J  may  be  reduced  to  Hths,  and  by  the  exer- 
cise of  our  reason  we  see  that  it  is  equal  to  |,  which  taken  from 
-|  leaves  J;  this  completes  the  work  with  the  fractions,  and  we 
have  but  to  find  the  difference  between  the  whole  numbers  as 
in  simple  subtraction. 

(3.)     What  is  the  difference    between  371    and  12/j-  ? 

Ans.  24*  f, 

OPERATION.  Emanation.    By  inspec" 

72  )  QQ  tion  we  here  see  that  the 

37|  =  27  j  fractions  belonging  to  the 

•J9JL-  56  whole    numbers   are   not 

^ of  the  same  denomina 

and  that  neither  can  be 

^-n3'  reduced  to  an  equivalent 

fraction  of  the  same  term 


100        Arithmetical  Exercises  and  Examples. 

as  the  other,  and  therefore  we  must  reduce  both  to  equivalent 
fractions  having  a  common  denominator,  before  we  can  sub- 
tract;  and  by  the  exercise  of  our  reason  we  see  that  72  is  the 
smallest  number  to  which  both  can  be  reduced,  which,  for  con- 
venience, we  set  below  the  fractions,  and  by  the  same  reason- 
ing as  given  in  the  preceding  examples  we  see  that  J  are  equal 

and  that  |  are  equal  to  |J,  which,  for  convenience,  we 
carry  to  the  right  of  the  respective  tractions,  and  to  economise 
time  we  set  only  the  numerators.  We  now  observe  that  the 
upper  traction,  belonging  to  the  greater  number,  is  less  than  the 
lower  fraction,  In-longing  to  the  lesser  number.  Therefore, 
before  \ve  can  subtract  the  fractions,  we  must  add  1,  reduced 
to  72ds,  to  27,.  which  We  now  take  C'£  from  i!1;!  and 

have  a  remainder  iie  fractional  part  of  our  answer.  We 

now  add  1  to  the  subtrahend,  because  we  previously  added  1  to 
thi-  minuend,  making  it  !.'»,  which  we  -ubtract  from  37  and 

t  remainder  ot  'J  I.  wi.  1  com- 

plete the  operation. 

4.  What  is  tin-  di£  n   ]  \  and  i!  ?     An- 

.">.  What  i>  tin1  dit  \> 

(I.  What  is  tlu-  difiL-ivinv  U-twren  .")-  and  !U  ?    An 

7.  What  is  th<-  diffrivn.v  bi-t \\von  7  and  o^.-  ?   Ans. 

8.  What  is  the  diffi  n  --»y  and  14?  An- 
What  is  the  diflxTriii-i-  l»i-t\v.  en  tin1  following  niun' 
!).      •:•  and  ;*.  Ans. 

10.  1-j  and  f  Ans. 

11.  'j  and   , 
1'2.      ^  and 

13.  [ian.i 

14.  ;,4,r  and  , 

15.  i^|and  ! 
Hi.  Hi  and  ^. 

17.      A  of  |-  and  i  of 
and  ;17-. 

•J«».      .T) |  and  U^. 

lid    3-vyl-. 

•  iV  • 

75  4  and  4.'  Ans 


Subtraction  of  Fractions.  101 

24.  31J-  and  17|.  Ans.   13|. 

25.  From  (>l  of  T4¥+13£  take  J-  of  }  of  15^-f'of  £. 

Ans.   13f|. 

2(5.      From  8J+6f—  ^  take  J  of  |  of  3  of  lf+21,. 

Ans. 


27.      From  t7}+f  take  6}—  f.  Ans.   12^. 

88.  <i.  K.  Shotwell  had  $38};  he  gave  S2i-  for  a  pair 
of  Indian  clubs.  S.Vj  for  books,  $1}  for  a  drawing  board. 
and  $|  for  ink  and  pencils.  How  much  had  he  left  ? 

Ans.   SJ- 

20.  \V.  A.  Weaver  had  67  .1  and  his  friend  gave  him  §.] 
more  ;  A.  Denis  had  $16o  and  he  spent  $5i  ;  How  much 
more  has  A.  Denis  than  W.  A.  Weaver?  Ans.  SL^. 

'.]().  E.  Schwartz  bought  2  bags  of  coffee  each  weighing 
lO.'J-l  pounds;  he  sold  27  I  pounds,  50f  pounds,  871  1'Ound.s 
and  45  \  pounds;  how*  many  pounds  has  he  left? 

Ans.    L16| 

31.  S.  Myers   bought    7.V.  gallons  of  molasses;   he  used 
4].  lost  by  l«ak.iu«i  2f,    and  so,d  22  ,:  gallons.      I  low  miu-li 
has  he  left?  Ans.   4(1'   gallons. 

32.  What  is  tlu«  ditli-n-nce  lu-tween  a  dozen  times  «>.  plus 
(J.j,  and  (J  tiiiHis  a  dozen  minus  one  dozen  and  a  halt".' 

Ans.   2  i 

J.  Famll  bi  'light  (\  chests  of  tea  weighing  3s. 
1  r,.:i!)',  and  43  i    pounds;  he  sold  120^  pounds  ;,nd 
used  64  pounds.      How  many  pounds  has  he  on  hand  ? 

Ans.    IL' 

34.  JY  Burba  owned  the  Steamer  Lsubel.  he  sold  ;^  ;  what 
is  -j  of  his  present  interest?  Ans.  -f^. 

3r>.  From  the  .-urn  oi  i)  1  and  S^  take  the  difference  of  14-2^ 
and  Of  Ans.  10^. 

36.  What  number  is  that  to  wliich  if  \(\\  be  added  the 
sum  will  be.  44^  ?  Ans.  27:. 


102      Arithmetical  Exercises  and  Examples. 

37.  C.  E.  McNeil  bought  £  of  §  of  a  vessel  and  sold  f 
of  3  of  his  share.  .  How  much  of  the  whole  vessel  has  he 
left?  Ans.  i. 

38.  E.  Brinkman  bought  a  barrel  of  molasses  containing 
H  I  gallons  ;   lie  sold  !) \  gallons  ;   how  many  gallons  remain 
in  the  barrel?  Ans.  31 J  gallons. 

•'V.>.  C.  GT.  De  Russy  bought  two  sacks  of  coffee  weighing 
respectively  1(11-1  and  lii:>:,:  pounds.  He  sold  to  J.  Walter 
ISO. I  pounds;  how  many  has  he  left? 

Ans.  138J  pounds. 

40.  A.  Buchanan  sold  to  J.  Bruns  J  of  «  of  his  planta- 
tion, what  part,  has  he  left?  Ans.  ^. 

41.  What  is  the  difference  between  •>  of  J  plus  5  and  % 
of  J  plus  J  ?  Ans.  Ty 

4±  .].  -I.  Srhonekas  and  M.  Shlenker  were  each  o  owner 
of  a  broom  and  brush  factory.  Schonekas  sold  J  of  his 
interest  to  E.  F.  Meyer,  and  then  •]  of  his  remaining  inter- 
est to  M.  Shlenker  who  subsequently  sold  %  of  f  of  his 
whole  interest  to  F.  Kranz.  What  is  the  present  interest 
of  each  owner  ?  Ans.  J.  J.  Slionekas  i  ;  E.  F.  Meyer  } ; 
M.  Shlenker  f|  and  F.  Kranz  £f 

43.  J.  J.  Manson  owned  J  of  the  Steamer  Natchez.  He 
sold  to  G.  Lindsey  ]  interest  in  the  Steamer,  and  to  W.  S. 
Keaghoy  .{  of  his  remaining  interest.  What  is  the  present 
interest  of  each  in  the  boat?  Ans.  Manson  f;  Lindsey  i 

and  Keaghey  J. 


Multiplication  of  Fr actions.  103 

MULTIPLICATION  OF  FRACTIONS. 

115.  Multiplication  of  fractions  is  the.proqess  of  multi- 
plying when  one  or  both  of  the  factors  contain  fractional 
numbers. 

In  the  multiplication  of  simple  numbers  we  saw  that  the 
result  of  multiplication  operations  was  increasing,  but  in  the 
multiplication  of  fractions,  when  the  multiplier  is  less  than 
a  unit,  the  result  is  decreasing.  This  is  evident  from  the 
fact  that  multiplication  is  the  process  of  repeating  the  mul- 
tiplicand as  many  times  as  there  are  units  in  the  multiplier, 
and  therefore,  when  the  multiplier  is  less  than  a  unit,  the 
multiplicand  will  be  repeated  only  a  }  art  of  a  time,  or  such 
a  part  of  itself  as  the  multiplier  is  p:\rt  of  a  unit. 

To  elucidate  the  principles  of  the  subject  and  render 
clear  the  reasoning  we  present  our  first  questions  in  prac- 
tical language  ;  and  to  aid  still  farther  in  comprehending 
the  work,  we  give  the  following  practical  definition  of  mul- 
tiplication. 

116.  MULTIPLICATION  is  that  operation   in  the  prac- 
tical business  computation  of  numbers  of  finding  the  cost  of 
either  a  part  of  one,  or  of  many  pounds,  yards,  barrels,  etc., 
when  we  have  the  cost  of  one  pound,  yard,  barrel,  etc.     On 
the  principle  or  fact  embraced  in  this  definition,  we  found  OUT 
reasoning  for  the  solution  of  every  question  tint  can   pos- 
sibly be  presented  in  multiplication,  either  of  simple  num- 
bers or  of  fractions. 

Considering  the  foregoing,  we  see  that  in  all  multiplica- 
tion questions  of  a  practical  nature,  we  must  necessarily 
reason  from  one,  or  unity ,  to  a  part  of  one  or  many.  Thus, 
if  1  pound  cost  50  /,  \  of  a  pound  will  cost  1th  part  of 
it ;  and  if  1  yard  costs  $2,  3  yards  will  cost  three  times  as 
much,  or  3  times  $2. 

In  the  solution  of  abstract  questions  we  apply  the  same 
system  of  reasoning  without  naming  the  factors,  and  thereby 
avoid  all  of  the  arbitrary  rules  given  in  the  Arithmetics  of 
the  day 


v  KM-         Arithmetical  Exercises  and  Examples. 

(1.)     What  will  41|  pounds  of  coflee  cost  at  21 \f  per 
pound  ?  Ans.  $8.97| 

OPERATION. 

j  Explanation.     In   this   example   by 

inspection    and    the   exercise    of  our 


reason,  we  see  that  we   have  in  the 


167  solution  both  increasing  and  decreas- 

-  ing  work  to  perform,  and  hence  to 

facilitate  the  operation  of  our  work, 
we  use  a  perpendicular  or  statement 
line,  on  the  right  hand  side  of  -which 
Ans.  we  piace  all  increasing  numbers,  and 
on  the  left  hand  side  all  decreasing 
numbers.  But,  be  it  remembered,  we  never  place  a  number  on 
either  side  without  giving  a  reason  therefor,  and  in  commencing 
the  solution  statement  of  any  problem  we  always  place  at  the 
top  right  hand  side  the  mimber  representing  the  article  or  thing 
to  be  increased  or  decreased,  or  that  which  the  conditions  of 
the  question  require  the  answer  to  be  in. 

By  further  inspection  and  reasoning  we  see  that  21  \f  are  to 
be  increased  41}  times,  and  hence  we  will  place  the  same  at  the 
top  and  on  the  increasingr  side  of  our  statement  line  ;  but,  before 
doing  so,  in  order  to  facilitate  the  work,  we  first  reduce  the  21J 
to  half  cents,  which  equals  423  cents,  the  denominator  of  which 
we  place  on  the  decreasing  side  and  the  numerator  on  the  in- 
creasing side  of  the  statement  line.  We  then  reason  as  follows: 
since  1  pound  cost  %3  cents,  J  of  a  pound  will  cost  i  part  of  it, 
and  as  this  conclusion  is  a  decreasing  one,  we  write  the  4  on 
the  decreasing  side  ;  then,  since  J  costs  the  result  of  the  state- 
ment thus  far  made,  1J7,  41f  reduced,  will  cost  167  times  as 
much,  which,  because  the  conclusion  is  an  increasing  one,  we 
write  on  the  increasing  side,  and  thus  complete  the  reason  and 
statement.  It  may  be  asked  how  we  know  that  if  1  pound  costs 
4j3^  J  of  a  pound  will  cost  J  part  of  it.  and  that  l  J7  will  cost 
lt>7  times  as  much.  We  answer,  by  the  exercise  of  our  reasoning 
faculties,  our  common  sense,  our  judgment,  which  is  the  only 
way  that  mortal  man  knows  anything. 

In  working  out  the  statement,  there  being  no  common  factors 
in  the  increasing  and  decreasing  numbers  that  can  be  cancelled, 
we  have  but  to  multiply  the  increasing  numbers  together,  which 
produce  7181,  and  the  decreasing  numbers  together,  which  pro- 
duce 8  ;  then  we  divide  the  7181  by  8  and  obtain  $8.97f  as  the 
result  of  the  reasoning  and  operation. 

In  all  simple  statements  the  result  is  always  of  the  same 
kind  or  character  as  represented  by  the  number  first  placed 
on  the  statement  lint 


Multiplication  of  Fractions.  105 

Multiply  £  by  f ,  or  to  express  the  problem  in  practical 
language : 

(2.)  What  will  I  of  a  yard  cost  at  -i  of  a  dollar  per 
yard?  Ans.  $}. 

OPERATION.  Explanation.     As   explained  in  the 

above  example,  by  inspection  and  rea- 


son we  see  that  the  i  of  a  dollar  is  the 
number  to  be  multiplied,  and  also  the 
number  representing  the  nature  of  the 


answer;  hence  we  first  place  the  same 
,  on  our  statement  line,  and  then  reason 

2  AnS.  ag  follows  :  if,  or  since,  1  yard  costs  | 

of  a  dollar,  J  of  a  yard  will  cost  $• 
part  of  it,  and  f-  wih  cost  5  times  as  much  as  J.  The  8  and  5 
are  placed  respectively  on  the  decreasing  and  increasing  sides 
of  the  statement  line,  because  the  reasoning,  when,  they  were 
respectively  used,  was  decreasing  and  increasing. 

In  working  out  the  statement,  we  first  cancel  the  5's,  and  then 
the  H  by  the  4,  and  thus  obiain  J  a  dollar  as  the  correct  result. 

The  reasoning  and  operation  of  the  foregoing  problems 
vi  1  solve  every  question  that  can  be  presented  in  multipli- 
«  ation  of  fractions. 

(3.)     What  will  58}  pounds  cost  at  in*/  per  pound? 


OPERATION.  I'.jjHiDKitinn.     The  reasoning  for  the 

tf  solution  of  this  problem  is  the  same 

25  as  that  given    in    the  first   example: 


"lit  39  hence  we  will  very  much  abridge  our 

explanation.     We    first    reduce    and 
\  place  on  the  line  the  IGj-V  ;  then  having 

$«r.75  Ans.      the  cost  of  1   pound,  we.  see  by  our 
reason  that  J  a  pound  will  cost  J  as 
much,  and  H7  pounds  will  cost  117  times  as  much  as  J. 

This  completes  the  statement,  and  in  working  the  same  we 
first  cancel  the  50  by  2,  then  the  117  by  the  3.  this  cancels  all 
of  the  decreasing  figures,  and  \ve  have  but  to  multiply  the  25 
by  39  and  produce  the  answer,  §9.75. 


106 


Arithmetical  Exercises  and  Examples. 


(4.)     What  will  3|  dozen  cost  at  $3|  per  dozen  ? 

OPERATION. 


17 


51 


Ans. 


F,.cplanation  For  reasons  above  giv- 
en, we  reduce  and  place  the  $3J  on 
tin-  line;  then  we  see  that  since  1 
di-y.-n  costs  $y,  J  of  a  dozen  will 
;  part  of  it,  and  ^  will  cost  15 
times  as  much. 


(5.)     What  will  5J  bushels  cost  at  15J/  per  pint? 

OPK'IATION.  l-'<f>l(in<ition.       By    inspection    and 

reason,  we  see  that  tin-  15£^  is  the 
number  to  he  increased;  hence  we 
reduce  and  place  the  same  on  the  line 
and  proceed  to  reason  as  follows:  if  1 
pint  COStfi  .  _  pints  or  a  quart  will 
cost  2  times  as  much,  and  if  1  quart 
costs  the  resuft  of  the  statement  now 
made,  H  quarts  or  a  peck  will  cost  8 
times  as  much,  and  if  a  peck  costs 
the  result  of  this  statement,  that  4 
pecks  or  a  bushel  will  cost  4  times  as  much,  and  if  a  bushel 
costs  the  result  of  this  statement,  that  J  of  a  bushel  will  cost 
J  part  of  it,  and  ^  will  cost  43  times  as  much. 


§53.32  Ans. 


(6.)     What  will  50}  pounds  of  tea  cost  at 
ounce? 


per 


OPERATION. 


f  21 

!  J(6 

(I20J 


2  Ans. 


Explanation.  Here  we  reduce 
and  place  the  10J^  on  the  line,  and 
reason  thus  :  since  1  ounce  costs 
^^,  16  ounces  or  1  pound  will  cost 
16  times  as  much,  and  since  1 
pound  costs  the  result  of  this 
statement,  £  of  a  pound  will  cost 
-  part  of  it.  and  w  will  cost  201 


times  as  much.     This  completes  the  reasoning  and  statement, 
which  worked  gives  $84.42,  answer. 


Multiplication  of  Fractions.  107 

(7.)     What  will  24  pounds  cost  at  9J/  per  pound? 

OPERATION. 

X  Explanation.     Reducing  and  plac- 
ing the  9J^  on  the  line,  we  reason 

f  iy  thus:  if  1  pound  costs  y%  24 
^  pounds  will  cost  24  times  as  ranch. 
12  This  completes  the  statement, 
which  worked  gives  $2.28,  the 

$2.28  An*.  answer' 

(8.)     What  will  14?  dozen  oost  at  §5  per  dozen? 

OPERATION. 

$  Explanation.     Placing  the  $5  on 

5  the    line,    we    reason    thus  :  if  1 

Q  A  A  dozen  co*ts  $5,  £  of  a  dozen   will 

cost  J  of  it,  and  </  will  cost  44 
times  as  much.  This  completes 
the  statement,  which  worked  out 
gives  $73£,  the  answer. 


$73i  Ans. 

(10.)     What  will  6  dozen  and  7  chickens  cost  at  $4.87} 
per  dozen?  Ans.  $32.09 £ 

OPERATION.  Explanation.      Reducing    and 

6  placing  the  $4.8 7 £  on   the   line 

2    .975  we  reason   thus:  Since  1  dozen 


12 


79 


cost    9750    ]    chicken   will   cosf 
the  12th  part  and  7U  chickens  79 
times  as   much    as    I  ;  or  thus, 
$32.09f  Ans.        since  1   dozen   costs   9J5^,   6^ 
dozen  will  cost  (J^  times  as  raur:i. 

(11.)     What  will  42  pounds  and  11  ounces  of  butter  cost 
at  22  J/  per  pound  ?  Ans.  $9.60£f 

OPERATION.  Explanation.      As     usual     we 

0  here    place    the    price    of  one 


2 
16 


A*  pound  on  the  statement  line  and 

£*  reason    as    follows        Since    1 

pound  cost  ^  1  ounce  will 
cost  the  16th  part  and  683 
Ans.  ounces  (which  is  42  pounds  and 


11  onnces)  will  cost  uss  times  as  much. 


108        Arithmetical  Exercises  and  Examples. 

117.      To  multiply  Abstract  Fractional  numbers. 

1.     Multiply  8J  by  3|. 

OPERATION.  Explanation. — In    this    problem    both 

;j  ->5  factors  are  abstract  numbers,  hence  we 

4  T£  5  cannot  give  the  same  analogical  reason- 

ing as  we  gave  in  the  foregoing  problems 
where  the  factors  were  denominate  num- 
125  bers  ;   although   were   we   to  do  so,   the 

result  so  far  as  the  figures  are  concerned, 
31}   Ans     would  be  correct.    We  therefore,  reduce 
and  place  the  8J,  the  number  to  be  mul- 
tiplied, on  the   statement  line  and  reason  as  follows  :  Since  1 
time  -2^5-   is  equal  to  -2S5,  J-  time    the  same  is  J  part  of  *j-  ami  Y 
are  15  times  as  many. 

MISri-LLANEOUS  EXAMPLES   IN    Mr LTII'LIOATION 
OF  FRACTIONS. 

1.      What  will  1(>  yards  cost  at  14:j^  per  yard? 

Ans.  82.31). 
'1.      What  will  23$  pounds  cost  at  35^  per  pound  ? 

Ans.  $s.:-ni. 

3.  What  will  J  of  a  yard  cost  at  $|  per  yard? 

"  Ans.   ^-|. 

4.  What  will  J  of  a  yard  cost  at  $J  per  yard? 

-     Ans.  $}. 
.">.     What  wrill  8^  pounds  cost  at  7JX  per  pound? 

Ans.  63J/. 

6.  What  will  10|  pounds  cost  at  9J^  per  pound  ? 

Ans.  99-j^/. 

7.  What  will  19f  pounds  cost  at  18|/  per  pound? 

Ans.  $3.60£|/- 

8.  What  will  25if  yards  cost  at  17 \ff  per  yard  ? 

Ans.  $4.55^. 

9.  What  will  11  f  yards  cost  at  12J/  per  yard? 

Ans.  $ 

10.  What  will  21}  yards  cost  at  16J/  per  yard? 

Ans.  $3.58-J. 

11.  What  will  14t  pounds  cost  at  12}^  per  pound? 

'Ans.   $1.74 A, 


Multiplication  of  Fractions.  109 

12.  What  will  31  J  pounds  cost  at  11J/  per  pound? 

Ans.  S3.46JJ. 

13.  Multiply  %\  by  12.  Ans.  5  J. 

14.  ,.  Multiply  If  by  13.  Ans.  10^. 

15.  Multiply  ^  by  19.  Ans.  10J-J-. 

16.  Multiply  7  by  f  Ans.  2f 

17.  Multiply  13  by  iV  Ans-  9iV 

18.  Multiply  105  by  ^.  Ans.  12. 

19.  Multiply  136  by  T3^.  Ans.  44-ff 

20.  Multiply  12  by  31|.  Ans.  382. 

21.  Multiply  25  by  3|.  Ans.  85. 
22  Multiply  19  by  T3r.  Ans.  5^. 

23.  Multiply  \\  by  if.  Ans.  }}. 

24.  Multiply  ||  by  f  Ans.  £|. 

25.  ailtiply  11  j  by  If.  Ans.  18f 
Multiply  2|  by  21  J.                               Ans.  50*. 

27.  Find  the  value  of  f  of  J  of  |  of  f  f  of  4. 

Ans.  A. 

28.  Multiply  7^  by  f  Ans.  6^-. 

29.  What  is  the  product  of  T97,  f  ,  f  'and  \  ? 

Ans.  ^. 

30.  What  is  the  product  of  If,  f  ,  2  and  5  J  ? 

Ans. 


31.  What  is  the  product  of  T\  of  2J  by  £  of  7J  ? 

Ans.  1H- 

32.  What  is  the  product  of  12}  multiplied  by  5}  times 
6|?  Ans'  464^. 

33.  At  i£  of  a  dollar  a  pound,  what  will  •£$  of  a  pound 
of  tea  cost?  Ans.  T9T  of  a  dollar. 

34.  What  will  51  dozen  buttons  cost  at  -g%  of  a  dollar 
per  dozen  ?  Ans.   J  of  a  dollar. 

35.  What  will  4}  yards  cost  at  4  J/  per  yard  ? 

Ans. 


110        Arithmetical  Exercises  and  Eaxmples. 

36.  What  will  9  J  yards  cost  at  9f  f  per  yard  ? 

Ans.  9: 

37.  What  will  12£  yards  cost  at  12 jy  per  yard? 

Ans.  $1.6C_ 

38.  What  will  12}  pounds  cost  at  12}/  per  pound? 

Ans.  $1.56}. 

39.  What  will  6}  pounds  cost  at  6}/  per  pound  ? 

Ans.  42T* 

40.  What  will  8f  pounds  cost  at  8J/  per  pound? 

Ans.  72|/. 

41.  What  will  19|  pounds  cost  at  19f/  per  pound? 

Ans.  $3.80££. 

42.  What  will  9}  pounds  cost  at  ll}/  per  pound  ? 

Ans.  $1.14^. 

43.  .  What  will  15}  pounds  cost  at  10}/  per  pound? 

Ans.  $1.62}. 

44.  What  will  40|  pounds  cost  at  22f  /  per  pound  ? 

Ans.  39.19^. 

45.  What  will  2812}  gallons  cost  at  $4.50  per  gallon  ? 

Ans.  $12656.25. 

46.  What  cost  471}  gallons  at  $3f  per  gallon  ? 

Am  $1592^-. 

47.  Sold  937852}  pounds  of  cotton  at  14|f  /  per  pound, 
what  did  it  amount  to  ?  Ans.  $135695.53f|. 

48.  If  a  man  earns  $2}  in  1  day  how  much  will  he  earn 
in  lr>}  days?  Ans.  $41}. 

49.  A  Contractor  pays  $1 }  per  day  for  labor  and  he  has 
370  men  employed  for  six  days.     How  much  money  will 
it  take  to  pay  them  ?  Ans.  $2775. 

50.  E.  J.  Denis  paid  ^j-  of  a  dollar  for  a  book  and  for 
paper  #  of  the  cost  of  the  book.     How  much  did  he  pay 
for  paper  ?  Ans.  60  cents. 

51.  Distillers  of  the  essence  of  rose  have  determined  by 
experience  that  it  requires  48000  pounds  of  rose  leaves  to 


Multiplication  of  Fractions.  Ill 

make  or  distill  one  pound  of  the  ottar  of  rose.  How  many 
pounds  of  rose  leaves  will  it  require  to  distill  50$  pounds 
of  the  ottar  of  rose  ?  Aus.  2442000  pounds. 

52.  If  a  pound  and  a  half  costs  a  cent  and  a  half  what 
will  25£  pounds  cost?  Ans.  25J  cents. 

53.  F.  Querens  Jr.  owned  i|  of  the  Steamer  Katie  and 
sold  f  of  his  share  to  G.  M.  Leahy,  what  part  of  the  whole 
Steamer  did  he  sell  ?  Ans.  f . 

54.  K.   E.   Terregrossa  can  work  the  problems  in  this 
book  in  4f  months,   how  many  months  would  it  take  him 
to  work  f  of  them  ?  Ans.  3^-  months. 

55.  E.  Schwartz  paid  $£  for  1  gallon  of  molasses,  what 
is  J  of  a  gallon  worth  at  the  same  rate?  Ans.  $|. 

56.  What  will  7?  boxes  of  raisins  cost  at  $2|  per  box  ? 

Ans.  $161. 

57.  iOn  one  occasion  at  the  New  Orleans  Opera  2  of  the 
ladies  and  gentlemen  present  were  French  ;   2  of  the  re- 
mainder Ameiican  ;  J  of  the  remainder  German,  and  the 
others  were  of  different    nationalities.     What    part  were 
Americans,  what  part  Germans  and  what  part  were  of  differ- 
ent nationalities  ?          Ans.   J  Americans,  TXT  Germans  and 

J  of  different  nationalities. 

68.  C.  Reynolds  owned  I  of  a  plantation  and  sold  %  of 
his  share  to  I).  C.  Williams,  who  sold  \  of  what  he  pur- 
chased to  E.  Szymanowski,  who  sold  J  of  what  he  pur- 
chased to  N.  Forcheimer.  What  is  Forcheimer's  share  in 
the  plantation  ?  Ans.  -^. 

59.  J.  Byrnes  owned  £  of  2000  acres  of  land  and  sold 
f  of  his  share  to  E.  H.  Wells,  who  sold  |  of  what  he  pur- 
chased to  H.  Clark.  How  many  acres  have  each? 

Ans.  J.  Byrnes  400  ;    E.  H.  Wells  450 ;  and  H.  Clark 
750  acres. 


112         Arithmetical  Exercises  and  Examples. 

DIVISION  OF  FRACTIONS. 

126.  Division  of  fractions  is  the  process  of  dividing 
when  the  divisor  or  dividend,  or  both,  contain  fractional 
numbers. 

In  the  division  of  simple  numbers  we  saw  that  the  result 
of  division  operations  was  decreasing,  but  in  the  division 
of  fractions,  when  the  divisor  is  less  than  a  unit,  the  result 
is  increasing.  This  iact  is  plain,  for  the  reason  that  the 
operation  of  division  is  the  process  of  finding  how  many 
times  the  dividend  is  equal  to  the  divisor,  and,  hence,  when 
the  divisor  is  less  than  1.  the  dividend  will  be  equal  to  the 
divisor  as  many  times  itself  as  the  divisor  is  part  of  1. 

In  practical  operations  we  usually  have  the  thive  follow- 
ing cases  or  questions  in  division  of  fractional  numbers. 

1st.  To  find  the  cost  of  mu*  pound,  yard,  or  trticle  of 
of  any  kind,  when  we  have  the  cost  of  many  pounds, 
yards  or  articles  of  any  kind  given. 

2d.  VTo/mt/  the  cost  of  one  pound,  yard  or  article  of 
any  kind,  when  we  have  the  cost  of  n  jxn-f  of  a  pound, 
yard  or  article  of  any  kind  given. 

3d.  To  find  the  number  of  pounds,  yards  or  articles 
of  any  kind  that  can  be  bought  with  a  specified  sum.  when 
we  have  the  price  of  one,  or  apart  of  one  pound,  yard  or 
article  of  any  kind  given. 

From  these  questions  we  see  that  division  is  the  converse 
of  multiplication  and  that  from  the  nature  of  the  question, 
we  must  reason  from  many  to  one  or  from  apart  of  one  to 
one.  Thus :  1st.  if  5  pounds  cost  50^,  1  pound  will  cost 
the  ith  part  of  it ;  in  the  2d.  case,  if  J  of  a  yard  cost 
$2,  J-  of  a  yard  •will  cost  the  J  part  of  it,  and  -^ths,  or  a 
whole  yard,  will  cost  4  times  as  much ;  and  in  die  third 
case,  if  jfjt  buy  1  yard,  or  any  other  thing,  \ft  will  buy 
the  -^-th  part  of  it,  and  f ,  or  a  whole  cent,  will  buy  2 
times  as  much. 

For  the  full  reasoning,  for  this,  ease,  see  the  explanation 
of  the  2d  problem. 


Division  of  Fractions.  113 

(1.)    Bought  7J    pounds    of  sugar  for  78}/.     What 
was  the  cost  per  pound  ? 

OPERATION.  Explanation.     By  inspection  and 

ft  the  exercise  of  our  reasoning  facul- 

315  21  t*es>  we  see  th*^  as  the  78}^  are 

the  cost  of  7J  poui.ds.  it  must  be 
divided  by   7J  in  order  to   obtain 
the  cost  of  1  pound.    We  therefore, 
Ans.    reduce  and  place  the  78}  on  our 


statement  line.     In  all  division  operations,  in  order  to  facilitate 
the  work,  we  thus  place  the  number  to  be  divided. 

We  then  reason  from  mauy  to  1,  as  follows  :  since  -V5-  pounds, 
which  is  7J  reduced  to  halves,  cost  -S^p/,  one-half  a  pound  will 
cost  the  15th  part  of  it,  and  2  halves,  or  a  whole  pound,  will 
cost  2  times  as  much.  This  completes  the  reasoning  and  state- 
ment. The  15  and  2  are  placed  respectively  on  the  decreasing' 
and  increasing  sides  of  the  line,  for  the  reason  that  when  they 
were  used  the  conclusion  arrived  at  were  respectively  decreasing 
and  increasing. 

{%)  At  10  \fl  per  pound,  how  many  pounds  can*  be 
bought  for  $3.92f? 

OPERATION.  Explanation.      By  inspection 

-  lb  and  reason,  we  see  that  the  ques- 

1  tion    requires    puunds    for   the 

A  result  or  answer.     Therefore,  in 

•--   *o  all    of  the    reasoning   we   must 

either  increase  or  decrease 
pounds.  To  aid  in  rendering 
36i  lb  Ans.  the  solution  easily  understood, 
we  first  place  the  1  pound  that  cost  lOff  on  the  right  of  our 
statement  line,  and  reason  as  follows:  since  4^.  which  is  10}p 
reduced,  buy  1  pound,  £  of  a  cent  will  buy  the  4Jd  part,  and  4 
fourths  or  a  whole  cent  will  buy  4  times  as  much;  then,  since 
1  cent  will  buy  the  result  of  the  statement  now  made,  J  of  a 
cent  will  buy  J  part,  and  2-1-^3-p  will  buy  3139  times  as  much. 
This  completes  the  reasoning  and  statement. 

The  placing  of  the  1  pound  on  the  statement  line  may  be 
omitted  and  the  reasoning  given  in  the  same  manner  as  when 
t  is  thus  placed. 


114         Arithmetical  Exercises  and  Examples. 


At  $|  per  yard,  how  many  yards  can  we  buy  for 


? 


OPERATION. 

Y 

1 

34 


Explanation.  For  reasons  given 
in  preceding  examples,  we  first 
place  1  yard  on  our  line,  and  then 
reason  as  follows:  since  J  of  a  dol- 
lar buy  1  yard,  \  will  buy  the  3d 
part,  and  J  or  a  whole  dollar,  4 
times  as  much:  then,  sin  (ye  1  dollar 
6|l-J-  yard,  Ans.  will  buy  the  result  of  our  state- 

ment. J  of  a   dollar   will  buy   the   8th   part,  and  J,   7  times  as 
much.     This  completes  the  reasoning  and  statement. 

V-l.)     At  $l-g-  JUT  puuml,  ho\v  many  pounds  can  we  buy 


OI'KKATION. 
ft 

A 

38 


H)0 


It)  Ans. 


Having  placed  1 
pound  on  our  line,  we  reason  thus: 
>in-:e  7  dollars  buy  1  pound,  £  will 
buy  the  7th  part,  and  |,  or  a  whole 
dollar,  f>  times  as  much,  and  38 
dollars  will  buy  38  times  as  much 
as  $1.  This  completes  the  reason- 
ing and  state.meut. 


(5.)     At  So  per  dozen,  now  many  dozen  can  we   buy 
for  V  i     ' 


Ol'K.r.ATlON. 

Duz. 

1 

I 


9Q1  ,        . 
484  doz  Ans. 


Explanation.  We  place  1  dozen, 
the  equivalent  of  $3,  on  the  line, 
and  reason  thus  ;  if  $3  buy  1 
dozen,  $1  will  buy  the  3rd  part; 
then,  if  $1  buys  the  result  of  the 
statement  now  made,  J  of  a  dollar 

will   buy  the  8th   part,   and  $6JS 
will  bliy  675  times  as  manv> 


Division  of  Fractions. 


115 


1  o 


(6.)     Bought  9f  yards  for  $22i.     What  was  the  price 
per  yard  ? 

Explanation.    For  reasons  given 
in  the  first  example  of  division, 
we  reduce  and  place  the  $22J  on 
4  our  line,  and  then  reason  thus  : 

if  '••f  yards   cost  V   dollars,  £  of 

a  yard  will  cost  the  75th  part, 
and  |  or  a  whole  yard  will  cost 
8  times  as  much.  This  completes 
Ans.  the  reasoning  and  statement.  In 
working  out  the  statement,  we  first  cancel  the  75  and  45  by  15  ; 
then  the  8  by  the  2  ;  then  we  multiply  the  3  and  4  together  and 
divide  the  result  by  the  5. 

(7.)     Bought  f  of  a  pound  for  30/.     What  was  the 
price  per  pound  ? 

OPERATION.  Explanation.    Placing  the  cost  on 

the  line,  we  reason  thus  ;  if  f  of  a 
%(h   1ft  pound  cost  30f,  £  will  cost  the  3d 

I   rr  part,  and  f  or  a  whole  pound  will 

cost    4    times    as    much  ;    which 
worked  out  gives  40f  or  the  cost 
40/   Ans.      of  !  pound. 
(8.)     Bought  5  boxes  of  indse.  for  SSI  J.     What  was  the 
price  per  box  ? 

OPERATION. 


Ans. 


65 


Explanation  1  lie  ^81|  being  the 
number  to  be  divided,  we  reduce 
and  place  the  same  on  the  line  ; 
then  reason  thus  :  if  5  boxes  cost 
3J5  dollars,  1  box  will  cost  the  5th 


EXAMPLES. 


(9.)     Bought  18|  pounds  for  37}/.     What  was  the 
price  per  pound  ?  Ans.  2^. 

10.  At   6i/  per  pound,   how  many   pounds   can   be 
bought  for  96IX  ?  Ans.  15ff 

11.  Bought  250J  dozen  for  $1251f     What  was  the 
price  per  dozen  ?  Ans.  $4.99f£f  . 


116         Arithmetical  Exercises  and  Examples. 


11 


12.  At  79-j$r/  per  pound,  how  many  pounds  can  bo 
bought  for  7287}/ ?  Ans.  91|-J2|. 

13.  At  $i  a  piece,  how  many  chickens  can  be  bought 
for  $25J  ?  Ans.  51. 

127.  DIVISION  OP  ABSTRACT  NUMBERS. 

(1.)     Divide  22}  by  5}. 

jo]  Explanation.     The  real  question 

to  be  determined  in  this  example 
is,  how  many  times  is  22J  equal  to 
5J,  and  as  both  the  dividend  and 

22  divisor  are  abstract  numbers,  we 

cannot,  therefore,  logically  reason 
~  .  as  in  the  preceding  problems,  and 

,    -„    ^ns-  accordingly     proceed    as    follows: 

the  22J  being  the  number  to  be  divided,  we  first  reduce  and 
place  the  same  on  our  statement  line;  then  by  inspection  and 
the  exercise  of  our  reason,  we  see  ihat  22J  is  equal  to  1,  22f 
times,  or  reduced  that  -^  are  equal  to  1,  ^-  times ;  and  if  equal 
to  I,  -M-  times,  it  is  equal  to  ^,  twice  as  many  times,  and  to  ty 
instead  of  J,  the  TJT  part.  This  completes  the  reasoning  and 
statement  of  the  problem,  and  the  same  character  of  reasoning 
and  statement  will  solve  all  division  problems  in  abstract 
numbers. 

(2.)     Divide  T^  by  i. 

OPERATION.  Explanation.     In    this    example 


15 

3 


2  the  dividend  being  less   than   the 


4 


divisor,  the  question  is,  what  part 
of  a   time  is    the    -f5   equal    to   J. 
Placing  the  •£§  on  the  line,  we  fea- 
Ans.  son,  as  in  the  above  example,  thus: 

o  are  equal  icTl,  ^  of  a  time ;  and  if  equal  to  1,  -fa  times,  it 
iV  equal  to  J,  4  times  as  many  times,  and  to  }  instead  of  J,  the 
3d  part. 


Division  of  Fractions.  117 

(3.)     Divide  3  by  } . 

OPERATION. 

;;  Explanation.     We  first  place  the 

*;  «j  3  on  the  line,  and  reason  thus  :  3 

is  equal  to   1,   3  times,   and   to  J 
instead    of   1,    3    times    as    many 
times  ;  and  to  -f  instead  of  £,  the 
J  part. 
Ans. 

i  1.)     Divide  14f  by  9. 

OPERATION.  Explanation.     Placing  the  num- 

ber to  be  divided  on  the  line,   we 
reason   thus  :  ^-  are   equal  to    1, 
-7^2-  times,  and  to  9  instead  of  1,  the 
9th  part. 
Ans. 


8 


The  solution  of  the  4  preceding  problems  elucidates  the 
only  correct  reasoning  for  dividing  abstract  fractional  num- 
bers. But  for  practical  work  we  would  not  advise  a  change 
from  the  reasoning  given  where  the  numbers  are  (^nomi- 
nate. 

MISCELLANEOUS     EXAMPLES    IX    DIVISION    OF    FRAC- 
TIONS. 

1.  Bought  4  yards  for  §143,  what  was  the  cost  per 
yard?  Ans.  $3f. 

2.  Sold  8J.  pounds  for  $1.87,  what  was  the  price  per 
pound  ?  Ans.  22  cents. 

3.  Paid  37J  cents  for  6}  yards  of  calico,  what  was  the 
price  per  yard  ?  Ans.  6  cents. 

4.  At  $1|  per  gallon,  how  many  gallons  can  be  bought 
for  $148.1  ?  Ans.   108  gallons. 

5.  Divide  ^  by  2.  Ans.  f. 

6.  Divide  ^9T  by  3.  Ans.  f 

7.  Divide  |f  by  5.  Ans.  ||. 

8.  Divide  ^  by  5.  Ans.  -^-. 

9.  Divide  7i  by  9.  Ans.  £ 
10.     Divide  2  by  f                                          Ans.  2J. 


118      Arithmetical  Exercises  and  Examples. 

11.  Divide  3  by  JT.  Ans.  7. 

12.  Divide  5  by  jf  Ans.  5 A. 

13.  Divide  21  by  ^-.  Ans.  33. 

14.  Divide  105  by  jf  Ans.  119. 

15.  Divide  fj-  by  ?55,  Ans.  4. 

16.  Divide  ft  by  £.  Ans.  12. 

17.  Divide  2i  by  jj.  Ans.  3. 
PT87J  Divide  ff  by  T4T.  Ans. 

— TOT    Divide  -5A  by  21.  Ans. 

20.  Divide  $  of  |f  by  f  Ans. 

21.  If  one  pound  of  tea  cost  £  of  a  dollar,  how  many 
pounds  can  be  bought  for  $25  ?  Ans.  30  Ibs. 

22.  Six  barrel*  of  flour  were  divided  among  some  poor 
families  in  such  a  manner  that  each  received  f  of  a  barrel; 
how  many  families  were  there?  Ans.  9. 

23.  If  a  boy  can  earn  T7T  of  a  dollar  in  one  day ;  how 
many  days  will  it  take  him  to  earn  £21  ?  Ans.  33  days. 

24.  Henry  walked   25  miles,  which  was  |  of  the  dis- 
tance Robert  walked  ;  how  many  miles  did  Kobert  walk  ? 

Ans.  30  miles. 

25.  At  the  battle  of  Germantown  the  British  lost  about 
600  men  ;  this  was  f  of  the   number  lost  by  the  Ameri- 
cans ;  and   the  number  lost  by  the   Americans  was  f  of 
the   number  they  received   as  re-enforcements  just  before 
the  battle.     How  many  men  did  the  Americans  lose,  and 
how  many  receive  as  re-enforcements  ? 

Ans.   1000  men  lost,  2500  re-enforcements. 

26.  A  man  had  his  store  insured  for  $9000,  which  was 
f  of  T9T  of  its  value  ;  what  was  the  store  worth  ? 

Ans.  $12375. 

27.  Sulphur  will  fuse  at  232°  Fahrenheit.     This  is  7} 
times   the  temperature  required  to   melt  ice.       At  what 
temperature  will  ice  melt  ?  4ns.  32°. 

28.  A  quantity  of  mercury  weighed  32062J  Ibs.,  which 
is  13i  times  the  weight  of  an  equal  bulk  of  water.     What 
would  an  equal  bulk  of  water  weigh  ?        Ans.  2375  ft>s. 

29.  A  pound  of  water  at  212°   F.  was    mixed  with  a 


Division  of  Fractions.  119 

pound  of  powdered  ice  at  32°.  The  united  temperature 
of  the  two  was  4T9g  times  the  temperature  of  the  mixture 
when  the  ice  became  melted.  What  was  the  temperature 
of  the  two  pounds  after  the  ice  became  melted  ? 

Ans.  52°. 

30.lx^VVhen  the  air  was  at  the  freezing  point,  a  cannon 

27(>KJ-S    feet   distant  from   New   Orleans  was   discharged. 

25  J  'seconds   elapsed  after  the  discharge  before  the  sound 

Breached  New  Orleans.     How  many  feet  per  second  did  the 

sound  travel  ?  Ans.   1090  feet. 

31.     Divido  2S7]  by  5.  Ans.  57 


Operation  without  the          Kjplanution.      We     first    divide 

line  statement.  ^  28T  b7   th«  Process   of  short 

r   .,,0^3  division  and  obtain  a  quotient  of 

57    and    a    remainder    of    2  ;    this 

remainder  we  reduce  to  a  fraction 

57^-Q-  Ans.       whose  denominator  is  the  same  as 

that  of  the  fraction  to  be  divided,  add  it  to  this  fraction  and 

then  divide  the  sum  by  5  and  annex  the  result  to  the  quotient 

57.     Thus  2=|-f  J=*fS  and  ^-5-6=4$. 

32.  Divide  1471  -,*.  by  !».  Ans.   163T^. 

33.  Divide  1044$  by  12\  Ans.  87TV 

34.  E.  T.  Churchill  divided  14T7^  dozen  apples  among  3 
boys  and  2  girls  ;  he  gave  each  girl  twice  as  many  as  each 
boy.     How  many  did  each  boy  and  each  girl  receive  ? 

Ans.  2r^  doz.  each  boy,  4i  doz.  each  girl. 

35.  Divide  1  by  \.  Ans.  5. 

36.  Divide  \  by  1.  Ans.  \. 

8J       A-        DJ        i 

37.  Divide  —  of  —  by  —  of  -  Ans.  f 

6J       H        4$       * 

38.  If  4J  pounds  of  coffee  cost  90  cents,  what  will  22  if 
pounds  cost?  Ans.  $4.55. 

39.  R.  E.  L.  Fleming  owns  1  of  the  capital  stock  of  a 
factory  valued  at  $24000;  he  gives   \  of  i  to  educational 
societies,  and  the  remainder  he  divides  equally  between  his 


120         Arithmetical  Exercises  and  Examples. 

four  children.     How  much  does  he  give  to  educational 
societies  and  how  much  does  each  child  receive  ? 

Ans.  $1500  to  educational  societies. 
$1875  each  child  receives. 

40.  C.  Craft  has  65}  yards  of  cloth,  2  yards  wide,  how 
many  yards  of  lining  if  of  a  yard  wide  will  be  required  to 
line  it?  Ans.   196}  yards. 

41.  Divide  IS  oranges  between  A.  and  B.   so  that  A. 
wtllhave  i  more  than  B.     What  number  will  each  have  ? 

Ans.  A.  10  ;  B.  8. 

42.  Divide  18  oranges  between  A.  and  B.  so  that  A. 
will  have  }  less  than  B.     What  number  will  each  have  ? 

Ans.  A.  7f ;  B.  lOf 

43.  A..  B.  and  C.   are  to  receive  $26   in  proportion  to 
,},  }  and  J.     What  will  each  receive  ? 

Ans.  A.  $12  ;  B.  $8  ;  C.  $ii. 

44.  A.  and  B.  can  do  a  piece  of  work  in  10  days  ;  A. 
alone  can  do   it  in    15  days.     How  many  days  will  it  take 
B.  to  do  it  ?  Ans.  30  days. 

45.  A.  and  B.  can  do  a  piece  of  work  in   14  days.     A. 
can  do  f  as  much  as  B,     How  many  days  will  it  take  each 
to  do  it,  work  in  IT  alone  ?  Ans.  24}  days  for  B. 

32|  days  for  A. 

46.  Three  persons,  A.,  B.  and  C.,  do  a  piece  of  work; 
A.  and  B.  together  do  -J-  of  it,  and  B.  and  0.  do  ^  ot  it. 
What  part  of  the  work  is  done  by  B  ?  Ans.  |~|-. 

-IT.  A  planter  remits  his  factor  $500  to  invest  in  rice 
and  coffee,  in  equal  sums.  He  pays  9i^  per  pound  for 
rice,  and  23|^  per  pound  for  coffee.  How  many  pounds 
of  each  did  he  purchase  ? 

Ans.  2702|-£  ft>  rice.     1069^  ft>  coffee. 

48:  W.  Quintel  has  $100  :  he  gives  f  of  it  for  five  bar- 
rels of  flour,  and  J  of  the  remainder  tor  4  barrels  of  pota- 
toes, and  with  the  remainder  he  buys  coffee  at  20^  per 
pound.  How  much  coffee  did  he  buy  ? 

Ans.  200  Ib  coffee. 

49.     C.  Wehrmann  owned  -^  of  a  stock   of  goods  :  he 


Division  of  Fractions.  121 

sold  -J-  of  his  share  for  $5000,  and  }  of  the  remainder  for 
$5000 .  and  then  the  balance  of  his  interst  for  $15000. 
What  part  did  he  sell  the  last  time,  and  what  would  the  whole 
stock  be  worth  at  that  rate  ? 

Ans.  ||  sold  last. 

$27777-J  value  of  stock. 

50.     What  quantity,   from   which   if  you  subtract   f  of 
itself,  the  remainder  will  be  15?  Ans.  24. 

148.— MISCELLANEOUS     EXAMPLES,      INVOLVING     THE 
PRINCIPLES  OF  ADDITION,  SUBTRACTION,  MULTI- 
PLICATION AND  DIVISION  OF    FRACTIONS. 

1.  Find  the  difference  between  ^  and  §,  $  and  ^,  ?  and 
T7T,  3|  and  2f ,  4§  and  }  of  3J  ?  Ans.  to  last,  3. 

2.  Find  the  sum  of  f  of  ^  and  £  of  /T.     Ans.  ££. 

3.  To  the  quotient  of  2f  divided  by  5£,  add   the  quo- 
tient of  3 J  divided  by  ^-r.  Ans.  7  £ . 

4.  A  number  was  divided  by  £,  and  gave  a  quotient  of 
20,  what  was  the  number?  Ans.   15. 

5.  What  number  is  that,  which  being  multiplied  by  ^r 
gives  as  a  product  ^  ?  Ans.  f . 

G.     What  number  is  that,  from   which,  if  you  take  f  of 
itself,  the  remainder  will  be  12?  Ans.  30. 

7.  What  number  is   that,   to   which,   if  you  add  -|  of 
itself,  the  sum  will  be  40  ?  Ans.  25. 

8.  A.  owns  f  of  a  store  which  is  worth  $25000.     He 
sells   |-  of    his  share ;  what   part  does   he   still   own,   and 
what  is  it  worth  ?  Ans.  owns  y1^,  worth  $2500. 

9.  $mith  owns  y5r  of  a  cotton  mill  and  sells  T3g-  of  his 
share  to  Jones  for  $33000  ;  what  is  the  mill  worth  at  that 
rate?  Ans.  $242000. 

10.  John  has  5  cents,    and   James   J    of  8  cents  ;  what 
part  of  James'  money  is  John's?  Ans.  f. 

11.  One   planter  raised  500    bales    of  cotton,    another 
raised  250  ;  what  part  of  the  first  one's  crop  is  the  second  ? 

Ans.  i. 


\'1'1        Arithmetical  Exercises  and  Examples. 

12.  The  sum   of  four  fractions  is  1;}.       Three  of  the 
fractious  are  ^,  i  and  f  ;  what  is  the  fourth?     Aus.  -f1. 

13.  What  number  is  that,  to  which  it  ^  of  f  £  of  1^-  be 
added,  the  sum  will  be  1  j  ?  Ans.  1. 

14.  Two  boys  bought  a  bushel   of  oranges,  one  paying 
2J  dollars  and  the  other  4§  dollars  :  what  part  of  it  should 
each  have?  Ans.  first,  ^f  ;  second,  f*. 

15.  A  farmer  sold  -£  of  his  mules  on  Monday  ;  on  Tues- 
day lie  bought  |  as  many  as  he  sold,  and  then  had  40  ;  how 
many  mules  had  he  at  first?  Ans.  f>(>. 

16.  F.  Gernon  gave  £,  i  and  k  of  his  money  to  different 
benevolent  institutions,  and  had  $1000  left,     How  much 
had  heat  first?  Ans.   S^IMMM). 

17.  J.  D.   Bothick  owning  -fa  of  a  rice  mill,  sold  §  of 
his  share  for  $8800.     What  was  the  value  of  the  mill? 

Ans.  SLM200. 

18.  A  book-keeper  worked  !>1  \  days,  and  after  paying  ii 
of  f  of  his  earnings  for  board  and  washing,   had  $1 
inaiiiing.     How  many  dollars  did  he  receive  in  all,  and  how 
many  per  day  ?  Ans.  $730  in  all,  $3  per  day. 

19.  A  planter  gave  50  bales  of  cott  >n  at  $50^  per  bale 
for  flour  at  $75  per  barrel.     How  many  barrels  of  flour  did 
he  receive  ?  Ans.  334. 

20.  Prophet  ^an  do  a  piece  of  work  in  6,  and  Fisher 
can  do  the  same  in  8  days ;  how  many  days  will  it    take 
both  together  to  do  the  work  ?  Ans.  3^  days. 

21.  Myers,  Levy  and  Hoffman  can  do  a  piece  of  work: 
in  10  days  ;  JJyers  and  Levy  can  do  it  in  15  days  ;  in  what 
time  can  Hoffman  do  it,  working  alone  ?    Ans.  30  days. 

22.  A  man  died  and  left  his  wife  $14400,  which  was  J 
of  f  |  of  his  estate.     At  her  death  she  left  £  of  her  share 
to    her    daughter.     How   many   dollars  did  the  daughter 
receive,  and  what  part  was  it  of  her  father's  estate  ? 

Ans.  $12000,  ff  of  her  father's  estate. 

23.  A  mule  and  dray  cost  $240;  the  mule  cost  If  times 
as  much  as  the  dray.     What  did  each  cost  ? 

Ans.  $150  mule,  $90  dray. 


Miscellaneous  Problems.  123 

24.  A  man  engaging  in  trade  lost  ^  of  the  money  he 
invested,  he  then  gained  $1000,  when  he  had  $3800  ;  what 
did  he  have  at  first,  and  what  was  his  loss? 

Ans.  #4900  at  first,  $2100  loss. 

25.  Forcheimer  lost  f  of  his  fish-line,  and  then  added 
25  J  feet  when  it  was  just  J  of  its  original  length.     What 
was  its  original  length  ?  Ans.  204  feet. 

26.  How  many  bushels  of  apples  at  $f  a  bushel,  will 
pay  for  ^  of  a  barrel  oranges  at  $6ir  a  barrel  ? 

Ans.  7J  bushels. 

27.  Sweeney  paid  %  of  his  year's  wages  for  board,  f  of 
the  remainder  for  clothes,  and  had  $80  left ;  how  many 
dollars  did  he  receive  for  labor  ?  Ans.  $560. 

28.  Purcell,  having  a  certain  number  of  cents,  gave  one- 
half  of  them  and  half  a  cent  over  to  one  beggar ;  one-half 
of  what  he  had  remaining  and  half  a  cent  ov>  r  to  a  second 
beggar ;  and  to  a  third,  one-half  of  what  he  then  had  and 
half  a  cent  over,  and  had  left  3  cents.     How  many  cents 
had  he  at  first?  Ans.  31  cents 

29.  Jol.n  lives  with   his    parents,    but  works   for   Mr. 
Smith  who  pays  him  $210  per  year.     His  parents  board 
him,  but  he  has  his  clothes  to  buy.     He  spends  ^  of  his 
wages  for  cigars,  -|  of  the  remainder  for  theater  tickets,  £ 
of  the  remainder  for  wine,  J  of  what  he  tfren  has  for  nov- 
els.    How  much  has  he  remaining  at  the  end  of  the  year 
to  pay  for  his  clothes  ?  Ans.  $30. 
*30     Joseph  worked  on  the  same  conditions  as  John.  He 
gave  ^  of  his  wages  to  the  cause  of  charity,  fa  of  the 
remainder  for  useful  books,  i  of  the  remainder  to  be  taught 
evenings,  paid  $100  for  clothes,  and  deposited  the  balance 
in  the  bank.     How  many  dollars  did  he  put  in  the  bank? 

Ans.  $50. 

31.  W.  T.  Harris  and  C.  E.  Jones  have  $1899,  Jones 
has  3J  times  as  much  as  Harris ;  how  much  has  each  ? 

Ans.  Harris  $422,  and  Jones  $1477. 

32.  J.  C.  Beals  can  solve  25  problems  in  50  minutes  and 


124        Arithmetical  Exercises  and  Examples. 

H.  H.   Barlow  can  solve  them  in  30   minutes.     In  what 
time    can  both  solve  them  ?  Ans.   18:]-  minutes. 

33.  E.  Meyer  purchased  200  barrels  of  flour  for  $1450 
and  sold   J  of  it  at  a  profit  of  $-1  per  barrel,  and  the  re- 
mainder at  $711Tr  per  barrel.     How  much  did  he  gain  ? 

Ans.  $67.50. 

34.  What  is  the  numerical  value  of 


Ans.    14V 

35.  M.  Ernst  bought  3S41  -I  pounds  of  cotton  at  7  ,:  pence 
per  pound;  what  did  it  cost  ?  Ans.  £1-4,  11.  (i1,  <1. 

36.  J.  W.    Anderson    has  3  dozen   oranges    which   he 
wishes  to  divide  between  Miss  Kate  and  Miss  Lucy,  so  that 
Miss   Kate  shall   receive   ]  more  than    Miss   Lucy.     How 
many  will  each  receive? 

Ans.   Mi>s  K.  2n  ami  Miss  L.  16. 

37.  A  tree  110   feet   high,   had  f  of  it  broken   off  in  a 
storm  ;  how  much  of  it  was  left  standing  •?    Ans.  44  feet. 

38.  What  cost  L'L'  J  pounds  of  coffee  at  21  J/  per  pound  ? 

Ans.  $4.94J|. 

39.  If  18?  yards  cost  $3.37  £  what  will  3J  yards  cost? 

Ans.  60  cents. 

40.  W.  D.  Maxwell  has   $600  of  which  he  wishes  to 
give  to  A.  J,   B.  J,  C.  ^   and  D.  i  ;  how  much   will   each 
receive?  Ans.  A.  $200,  B.  $150,  C.  §120 

and  D.  $100. 

41.  R.  L.  Paul  has  $600  which  he  wishes  to  give  to  A* 
B.,  C.  and  D.  in  the  proportion  of  J,  },  ^  and  i  ;  how 
much  will  each  receive?       Ans.  A.   $210j|,   B.   $157}f 

C.  $126^  and  D.  $105^. 

42.  C.  M.  Huber  and  A.  J.  Hohensee  bought  on  specu- 
lation $800   worth  of  merchandise,   of  which   Huber  paid 
$500  and  Hohensee  $300  ;  they  sold  to  W.  A.  Tomlinson 
i  of  the  whole  for  $400.     How  much  of  the  $400   must 
Huber  and  Hohensee  receive  respectively,  in  order  to  con- 
stitute each  J  owner  in  the  renriinder  of  the  goods  ? 

Ans.  Huber  $350  and  Hohensee  $50. 


Miscellaneous  Problems.  125 

43.  If  a  yard  and  a  half  cost  a  dollar  and  a  half  what 
will  twelve  and  a  half  yards  cost?  Ans.  $12J. 

44.  If  3  is  the  third  of  6  what  will  the  fourth  of  20  be? 

Ans.  7J. 

45.  Greo.  Meyer  owned  a  quantity  of  rice,  of  which  he 
sold  i  for  $99.60  ;  what  is  f  of  the  remainder  worth  at  the 
same  rate?  Ans.  $16.60. 

46.  F.  Miller  paid  $60  for  f  of  an  acre   of  land  ;  what 
is  the  value  of  f  of  an  acre  ?  A.ns.  $50. 

47.  S.  Benavides  bought  937852J  pounds  of  cotton  at 
1415^  per  pound  ;  what  was  the  cost? 

Ans.  $135695.53ff. 

48.  J.  Koch  invested  }  of  his  money  in  sugar,  J  in  rice, 
|  in  coffee  and  deposited  in  bank  $2645.     How  much  money 
had  he  at  first  ?  Ans.  $63480. 

49.  L.  Meyer  spends  i  of  his  time  in  study,  i  in  labor, 
i  in  rest  and  recreation,  and  the  remainder  in  sleep.     How 
many  of  the  24  hours  of  a  day  does  he  sleep? 

Ans.  7  hours. 

50.  A  loafer  spends  4  hours  per  day  sauntering  on  street 
corners,  3  hours  smoking   and   drinking,   i  of  the  day  in 
sleep,  i  of  the  day  in  drunkenness,  y1^  in   eating,   TV  in 
quarreling  and  the  remainder  of  the  day  in  gaming.     How 
many  hours  does  he  spend  in  guming  ?       Ans.  3  hours. 

51.  An  industrious  young  lady  spends  i  of  her  time  in 
the  performance  of  household  affairs,   i   in  reading  good 
books,  y1^  in  physical  exercise  in  the  open  air  and  sunlight, 
i  in  the  practice  of  music,  singing  and  parlor  amusements, 
or  social   intercourse,  2   hours  per   day   in  eating,  and  the 
remainder  of  the  day  in  sleeping.     How  many  hours  per 
day  does  she  devote  to  each  ? 

Ans.  6  hours  to  household  affairs;  4  hours  to  reading; 
2  hours  to  exercise  ;  3  hours  to  music,  etc.;  2  hours 
to  eating,  and  7  hours  to  sleep. 

52.  A  fashionable  young  lady  spends  £  of  her  time  in 
dressing,  painting  and  making  her  toilet,  J  in  reading  nov- 
els and  papers  of  senseless  fiction,  £  in  making  calls  and 


126         Arithmetical  Exercises  and  Examples. 

gossiping,  y1^-  in  street  promenading,  ^  in  criticising  indus- 
trious young  men,  and  speculating  upon  the  qualities  and 
fortune  of  an  anticipated  husband,  J?  in  making  remarks 
derogatory  to  the  ohanicti-r  of  those  who  labor,  while  her 
own  mother  is  perhaps  cooking  or  washing,  T^  in  enter- 
taining young  men,  and  the  remainder  in  eating  and  sleep- 
ing. How  many  hours  does  she  devote  to  useful  service, 
and  how  many  to  eating  and  sleeping  ? 

Ans.  0  hours  to  useful  service;  8  hours  to  eating  and 
sleeping. 

53.  A  man  willed  }  of  his  property  to  his  wife,  J  of  the 
remainder  to  his  daughter,  and  the  remainder  to  his  SOD; 
the  difference  between  his  wife  and  daughter's  share  was 
$8000.     How  much  did  he  give  his  son  ? 

Ans.  $4800. 

54.  R.  W.  Tyler  owned  a   J    interest  in  a  factory ;  he 
sold  to  C.  Modinger  \  of  his  interest  for  $15000.     What 
interest  does  he  still  own,  and  how  much  is  it  worth  at  the 
rate  received  for  the  part  sold  ? 

Anfe.  he  still  owns  f,  worth  $15000. 

55.  J.  Cassidy  owned  I  of  the  Steamer  R.  E.  Lee.     He 
sold  to  Gr.  Buesing  i  interest  in  the  Steamer  for  $20000  ; 
and  to  J.  C.  Beals  \  of  his  remaining  interest  at  the  same 
rate.     What  did  he  receive  for  the  last  sale,  and  what  is 
his  remaining  interest  in  the  boat  ? 

Ans.  he  received  $30000  ;  T9F  remain- 
ing interest. 

56.  N.  Puech  and  A.  Palacio  bought  on  joint  account 
each  J  the  New  Orleans  Cotton  Factory.     N.  Puech  sold  J 
of  his  interest  to  11.  Krone,  and   subsequently   J  of  his 
remaining  interest  to  A.  Palacio,  who  subsequently  sold  J 
of  I  of  his  whole  interest  to  R.   Lynd  for  $7500.     What 
is  the  factory  worth  at  the  same  rate,  and  what  is  each 
owner's  interest? 

Ans.  $32000  value  of  Factory ;  Puech  owns  J  ;  Krone 
i  ;  Palacio  |f ,  and  Lynd  -J-f . 

57.  L.  Kaiser  bought  f  of  f  of  28  J   barrels  of  apples, 


Miscellaneous  Problems.  127 

and  sold  to  S.  L.  Crawford  f  of  9  barrels  for  $20},  which 
was  $1.50  more  than  the  same  cost.  What  was  the  cost 
of  the  whole,  and  how  many  barrels  has  he  unsold  ? 

Ans.  $39^-  cost ;  7*S  barrels  unsold. 

58.  W.  D.«Maxwell  gives  |  of  his  annual  income  to  aid 
meritorious  young  men  in  obtaining  an  education;   J  of  the 
remainder  for  the  publication  and  free  distribution  of  books 
treating  of  the  awful  injury  to  the  human  race  by  the  use 
of  tobacco,  tea.  coffee  and  wine  ;  J  of  the  second  remainder 
for  various  benevolent  purposes.     The  balance  $5490  he 
retains  for  his  own  personal  use ;  how  much  does  he  give 
for  each  object  named  ? 

Ans.  $8235  for  meritorious  young  men  ;  $8235  for  the 
publication  and  distribution  of  books ;  and  $2745 
for  various  benevolent  purposes. 

59.  What  is  the  smallest  sura  of  money  for  which  I 
could   purchase   a  number  of  bushels  of  oats,  at  $-f^  a 
bushel;  a  number  of  bushels  of  corn,  at  $f  a  bushel ;  a 
number  of  bushels  of  rye,  at  $1 J  a  bushel ;  or  a  number  of 
bushels  of  wheat,  tit  $2|  a  bushel ;  and  how  many  bushels 
of  each  could  I  purchase  for  that  sum  ? 

Ans.  $22 '.  ;  72  bushels  of  oats ;  3G  bushels  of  corn  ; 
15  bushels  of  rye;  10  bushels  of  wheat. 

60.  There  is  an  island  15  miles  in  circuit,  around  which 
A.  can  travel  in  J  of  a  day,  B.  in  i  of  a  day,  and  a  horse 
car  in  •£$  of  a  day.     Supposing  all  to  start  together  from 
the  same  point  to  travel  around  it  in  the  same  direction,  * 
how  long  must  they  travel  before  coming  together  again  at 
the  place  of  departure,  and  how  many  miles  will  each  have 
traveled  ? 

Ans.  10}  days;  A.  210  miles;  B.  180  miles;  Horse 
Car  525  miles. 

DECIMAL   FRACTIONS. 

150.  A  Decimal  Fraction  is  one  whose  integral  unit  is 
divided  according  to  the  decimal  scale  ;  therefore  thedenomi- 


I'-.s        Arithmetical  Exercises  and  Examples. 

nator  is  some  power  of  ten  ;  as  10,  100,  1000,  etc.  The 
word  decimal  is  derived  from  the  Latin  word  decem,  which 
means  ten. 

151.  The  Decimal  Point  (.)  is  used  to  distinguish 
decimals  from   whole   numbers.     When   there  are    mixed 
numbers,   it    also   separates  th«>  whole   numbers   from    the 
decimals. 

The  following  are  decimal  fractions  :  y8^,  y1-^,  TW^, 
a"d  T^inr?;  tn°y  are  here  written  as  common  fractions, 
but  generally  the  denominator  of  decimal  fractions  is  omit- 
ted and  the  value  i,s  indicated  by  the  location  of  the  deci- 
mal point  before  the  numerator. 

To  write  these  fractions  according  to  the  decimal  nota- 
tion, they  would  be  written  thus  : 

T7^  decimally  expressed  is   . ,~. 

iW  Decimally  expressed  is  .If). 

yW<r  decimally  expressed  is  .137. 

T$Mo  decimally  express.-,!  is  .0123. 

152.  Notation  Of   Decimals.       Whenever  decimal 
fractions   are   expressed   decimally,    the    numerator    must 
have  as  many  decimal  places  as  there  are  naughts  in  the 
denominator.     Thus  TV=.4  ;  .yV^-16  5   .^WW=-1456- 
When  the  number  o*1  naughts  in  th"  denominator  is  greater 
than   the   number   of   figures   in    the  numerator,   naughts 
must  be  prefixed   to  the   numerator   until   the   number  of 
places  is  equal  to  the  naughts  in  t!:e  demominator.      Thus 
Y*T=-04;  TrtW=-°°7;  TTrVVm^-00125,  etc. 

When  the  number  of  naughts  in  the  denominator  is  less 
than  the  figures  in  the  numerator,  the  result  or  value  of 
the  fraction  will  embrace  a  whole  number  and  a  fraction. 

153.  A  Pure  or  Simple  Decimal  consists  of  a  dec- 
imal fraction,  decimally  expressed   or  written.     Thus  .5, 
.42,   .875   and    .1256    are    pure    decimals,    and    are    read 
respectively  5  tenths  ;  42  hundredths  ;  875  thousandths  and 
1250  ten  thousandths. 


Decimal  Fraction*.  129 

154.  A  Mixed  Decimal  consists  of  a  whole  number 
and  a  decimal.     Thus  24.5  and  41.25  are  mixed  decimals. 
They  are  read  respectively,   24  and   <>   tenths  ;  41  and  25 
hundredths. 

155.  A   Complex   Decimal   consists    of  a    decimal 
with  a  common   fraction  annexed.     Thus   .15J  and  .005i 
are   complex    decimals.     They   are   read  respectively,   15  J 
hundredths  ;  5i  thousandths. 

156.  A  Circulating  Decimal  is  one  in  which   a  fig- 
ure or  set  of  figures  constantly  repeats  itself.     Thus  J= 
.3333+,   |=.142857  +  ,   H=.7333o+.     The  figure  or 
set  of  figures  which  is  repeated  is  called  a  Repeteml.     If 
the  repetend   consists  of  only  one  figure,  a  dot  is  placed 
over  it ;  if  of  a  set  of  figures,  a  dot  i    placed  over  the  first 
and  last  figures,  as  J=.3,  J=.6,  j*f=. i  9, '<^=. 142857. 

157.  .A  Pure  Circulating  Dedmal  is  one  which 

contains    only    the    repetend  ;    as    $—.6,    \—.  142857,     % 


158.    A  Mixed  Circulating  Decimal  is  one  which 

contains  other  figures  than  the  repet  ;nd  ;  at  J=.  ie,  |jjj- 
—.647. 

There  are  still  other  kinds  of  circulating  decimals,  but 
as  they  are  of  very  little  practical  im  )ortauc3,  we  will  not 
here  consider  them. 

159.-  Decimal  fractions,  like  whole  numbers,  decrease 
towards  the  right  and  increase  towaids  the  left  in  a  ten- 
fold ratio,  and  hence  the  prefixing  of  laughts  between  the 
decimal  figures  and  the  decimal  poin,,  or  the  removal  of 
the  decimal  point  towards  the  left  diminishes  their  value 
ten- fold,  or  divides  the  decimal  by  tea  for  oach  order  or 
place  removed,  and  conversely  the  removal  ol  the  decimal 
point  to  the  right,  increases  the  value  ten- fold  or  multi- 
plies the  decimal  by  ten  for  each  place  retuov  -d. 

Annexing    naughts  "to   decimals  does  not  change  their 


130        Arithmetical  Exercises  and  Examples. 

value,    because    the    significant    figures    are    not    thereby 
removed  nearer  to  nor  farther  from  the  decimal  point. 

Decimal  orders  are  also  called  decimal  y/A/m<?.  each  order 
being  counted  as  one  place.  Thus  in  .0043  there  -are  four 
decimal  plates,  although  the  3  is  of  the  fifth  decfiii't/  /•/>!<•, 
from  unity,  the  base  of  the  system. 

The  following  table  will  illustrate  more  fully  the  relation 
of  whole  numbers  and  decimals,  with  their  incrcasim:  and 
decreasing  orders  to  the  left  and  right  of  the  decimal 
point. 

TABLE. 

WHOLE    NUMBERS.  DECIMALS. 

cc 

•TO 


3 

C3 

• 

.2 

14 

02 
^^                     -f 

s  I 

if 

Is     .j 

«*-  rr 

d 

•s.§  . 

en  HJ§    aa 

sl'sll     « 

.  GO'S  2      |§ 
&      l|  IF,:  I5? 

•73    s    o  ^  ^  -—  "^ 

^•SlJSflSj 

•"O 

.C    ^J            g  ns          jj 

g.g'Tj     g^^^o^^ 

~   ^ 

S  p  S  o  a  S  ^ 

g(3|gt2^5 

lllllwlll 

98 

7654321 

.23456789 

Orders  of  ascending  scale.  Orders  of  descending  scale 

This  number  is  read  987  million  654  thousand  321,  and 
23  million  456  thousand  789  hundred-millionths. 

In  order  to  clearly  understand  decimals,  we  must  bear 
in  mind  that  the  unit  one  is  the  basis  of  all  numbers,  inte- 
gral and  fractional,  abstract  and  denominate,  and  that  all 
mathematical  operations  have  this  fundamental  principle 
for  their  origin,  and  every  number  is  but  a  multiple,  either 
ascending  or  descending  of  unity  or  one. 

The  names  of  the  decimal  orders  are  derived  from  the 
names  of  the  orders  of  whole  numbers.  Thus  the  names 
of  the  orders  in  the  ascending  scale,  are,  after  units,  tr.ns, 


Decimal  Fractions.  131 

hundreds  etc.,  and  the  orders  in  the  decending  scale,  are, 
after  units,  tenths,  hundredth^  etc.,  the  decimal  orders 
being  the  reciprocal  of  the  orders  of  whole  numbers  equal- 
ly distant  with  themselves  from  the  u  iits. 

Numeration  of  Decimals.  In  reading  decimal  frac- 
tions the  entire  decimal  is  regarded  as  reduced  to  units  of 
the  lowest  order  expressed,  and  the  name  of  this  order  is 
given  to  the  entire  number  of  decimal  units.  Thus  .25  is 
read  twenty -five  hundredths. 

Before  reading  a  decimal,  we  must  determine  1st.  How 
many  units  are  expressed.  To  do  this,  we  numerate  and 
read  the  significant  figures  of  the  decimal  as  in  whole  num- 
bers. 2nd.  We  must  determine  the  name  of  the  lowest  order 
in  the  decimal.  To  do  this,  we  numerate  the  number  dec- 
imally. Thus  to  read  .001073,  we  c<  mmence  at  the  3  and 
numerate  to  the  1  thousand,  and  thus  find  that  1073  units 
are  expressed  ;  then  we  commence  ao  the  decimal  point 
and  numerate  decimally  to  the  3  and  thus  find  that  mil- 
lionths  is  the  lowest  order,  we  then  re  id  1073  millionths. 

160.  EXERCISES. 

Read  the  following  numbers : 

1.  16.008  ;  reads  thus,  sixteen  units  and  e  ght  thous- 
andths. 

2.  .94f ;  reads  thus,  ninety-four  and  three-eights  hund- 
reths. 

3.  5067.4005  ;  reads  thus,    5067  units  and  4005  ten- 
thousandths. 

4.  Write  and  read  197.8;  4.68907;  .00073;  48.769- 
146. 

5.  Write  and  read  2.491;  10.0101089167;  582.400- 
410905. 

6.  Write  and    read    5841. 291f;    8000.0000000217; 
9876541.1000001. 


132      Arithmetical  Exercises  and  Examples. 


161.  Writing  Decimals.  In  writing  decimals  we 
write  down  the  given  number  as  if  it  were  a  whole  nnmber ; 
then,  to  facilitate  the  operation,  we  numerate  from  right  to 
left,  beginning  the  numeration  with  tenth*,  and  continue 
until  we  come  to  the  required  place  or  order,  always  writing 
O's  to  fill  the  places  not  occupied  by  significant  figures. 
Thus,  to  write  25  ten  thousandths  we  first  write  the  2f>  ; 
then  we  begin  at  the  right  and  numerate  thus,  tenths,  hun- 
dredths,  thousandths,  ten  thousandths;  by  this  we  find  that 
four  places  are  required  and  as  there  are  but  two  figures  in 
the  number  we  prefix  two  O's  and  obtain  the  correct  result 
.0025. 

1.     Write  104  hundred  thousandths.      Ans.  .00104. 

Explanation.  According  to  the  above 
diiections  we  write  the  104  and  then 
commence  on  the  right  and  numerate 
thus;  tenths,  huudredths,  thousandths, 
ten  thousandths,  hundred  thousandths. 

This  numeration  shows  that  five  places  are  required  and  as  we 

have  but  tkrie  we  therefore  prefix  two  O's. 


OPERATION. 
.00104. 


162.  EXERCISES. 

1.     Write  10101  hundred  billionths. 

Ans. 


.00000010101. 


Write  decimally,  numerate  and  read  the  following : 


2 

314  millionths. 

6.     1205  ten  millionths. 

s! 

12  thousandths. 

7.       897  hundred  billionths. 

4. 

107  billionths. 

8.            1  isextillionth. 

5. 

1  trillionth. 

9.  21001  ten  vigintillionths. 

10. 

_5* 

14. 

60409 

17             87 

1  0 

100000000 

10000 

11. 

TOO 

748^ 

If 

15     — 

18         —  7 

12. 

TOOO 

1000000 

100 

1042J 

1  fi 

1 

-.  ^              990099 

i  3 

10.      1 

i  ii  ii  m:  !i  >MI  K  i 

*«'•      i<vwv  LI  infWKWtAfi 

J.O. 

100000 

Reduction  of  Decimals.  133 

163.  PRINCIPLES. 

From  the  foregoing  work  we  recapitulate  the  following 
principles  : 

1.  Decimals  are  governed  by  the  same  laws  of  notation 
as  whole  numbers,    hence  the  value  of  any  decimal  figure 
depends  upon  the  place  it  occupies. 

2.  Each  removal  of  the  decimal  point  one  place  to  the 
right  is  equivalent  to  multiplying  the  decimal  by  10. 

3.  Each  removal  of  the  decimal  point  one  place  to  the 
left  is  equivalent  to  dividing  the  decimal  by  10. 

4.  Annexing  or  rejecting  naughts  at  the  right  of  any 
decimal  does  not  change  its  value. 

REDUCTION  OF  DECIMALS. 

164.      To  reduce  Decimal  Fractions  to  a  common   de- 
nominator. 

1.     Reduce  .7,  .18,  .2581  and  .045  to  a  common  denom- 
inator. 

OPERATION. 

.7000  Explanation.     To  reduce   decimals  to 

1800  E  common  denominator  we  have  but  to 

'  annex  a  sufficient  number  of  O's  to  give 


each  decimal  the  same  number  of  places. 
.0450 

165.      70  reduce  a  decimal  to  a  common  fraction. 

1.  Reduce  .25  to  a  common  fraction. 

OPERATION  Explanation.     In  all  problems  of  this 

25  .     '  kind  we  simply  write  the  decimal  as  a 

1  00==4    A-ns-          common  fraction  and  then  reduce  it  to 
its  lowest  terms. 

2.  Reduce  .125  to  a  common  fraction. 

OPERATION. 

TTnfV^*  Ans- 

3.  Reduce  .59|  to  a  common  fraction. 

FIRST    OPERATION. 

59f      ±J£ 

—--d^^^-if  An, 


134       Arithmetical  Exercises  and  Examples. 

Explanation.  To  reduce  complex  decimals  to  simple  fractions 
we  first  write  the  decimal  as  a  common  fraction;  then  we  reduce 
both  the  numerator  and  denominator  to  the  fractional  unit  of 
the  denominator  contained  in  the  numerator  term  of  the  frac- 
tion, and  thus  obtain  a  complex  fraction,  which  we  reduce  to  a 
simple  fraction. 

SECOND   OPERATION.  Ksplanation.     Here,    when    redu- 

59|  cing  the  fraction   to  the  fractional 

47  5 19     A  ns       unit  of  the  denominator  contained 

10n~~  ^  in  thenumerator  term  of  the  frac- 

tion,   we    shorten     the     work    by 

omitting  the  denominator  (8)  in  both  terms  of  the  complex 
fraction,  and  writing  the  result  as  a  simple  fraction.  By  this 
process  we  save  the  operation  of  division,  the  result  of  which, 
is  the  cancelling  of  the  denominator  in  both  terms  of  the  com- 
plex fraction. 

Reduce  the  following  decimals  to  common  fractions : 


4. 

.8. 

Ans.  A. 

13. 

.88.  •       Ans. 

5. 

.05 

Ans.  Tf0-. 

14. 

.909.       Ans. 

6. 

.25. 

Ans. 

15. 

.00025.  Ans. 

7. 

.125. 

Ans. 

1G. 

.4«j.           Ans. 

8. 

.675, 

Ans. 

17. 

.055|.        Ans. 

9. 

.105. 

Ans. 

18. 

.008f     Ans. 

10. 

.07. 

Ans.  Tfo. 

19. 

.00054/s-.  Ans. 

11. 

.005. 

Ans.  rfa. 

20. 

.999.       Ans. 

12. 

.1045. 

Ans. 

21. 

4007  rV  Ans. 

167.      To    reduce  common  fractions  to  equivalent  deci- 
mals. 

1.     Reduce  f  to  a  decimal. 

OPERATION.  Explanation.    To  reduce  common  frac- 

8)3  000  tions  to  decimals,  we  annex  naughts  to 

_    '  the  numerator  and  divide  by  the  denom- 

inator,  then  point  off  as  many  places 
.375  Ans.  for  decimals  as  there  were  O's  annexed. 
When  a  remainder  continues  beyond  four  or  six  places,  we  dis- 
continue dividing  and  write  the  sign  -j-  to  the  right  of  the  last 
figure  obtained,  which  indicates  that  the  quotient  is  not  com- 
plete. The  annexing  of  O's  to  the  numerator  is  equivalent  to 
multiplying  it  by  10  for  each  naught  annexed,  consequently 
the  quotient  obtained  is  as  many  times  10  too  great  as  there 
were  O's  annexed  ;  and  hence  the  reason  for  pointing  off  as 


Reduction  of  Decimals.  135 

many  places  in  the  quotient  as  there  were   O's  annexed  to  the 
numerator. 

2.     Reduce  -f-  to  an  equivalent  decimal. 
OPERATION. 
7)5.000000 


.714285+  Ans. 
3.     Roduce  yl-j-  to  an  equivalent  decimal. 

FIRST  OPERATION.  SECOND  OPERATION. 

725)3.000000(4137+  725)3.   (.004137-f  Ans. 

2900         .004137+  Ans.      30  tenths. 

1000  300  hundredths. 

725 
3000  thousandths. 

2750  2900 

2175 

1000  ten-thousandths. 

5750  725 

2750  hundred  thousandths, 
2175 


5750  millionths. 
5075 

675  Remainder. 

Explanation.  Here,  in  the  first  operation,  we  annex  six  O's 
and  obtain  but  4  figures  in  the  quotient.  Therefore  in  order 
to  point  off  as  many  decimal  places  as  we  annexed  O's,  we  pre- 
fix two  O's  and  thus  obtain  the  correct  result.  The  reason  for 
this  will  appear  clear  if  we  consider  each  step  of  the  work  as 
performed  in  the  second  operation.  We  are  to  divide  or  meas- 
ure 3  by  725,  and  we  first  see  that  3  is  not  equal  to  725  any 
whole  or  unit  number  of  times  ;  we  therefore  write  the  decimal 
point  in  the  quotient,  annex  a  0  to  the  3  units  and  thus 
reduce  it  to  30  tenths,  which  we  also  see  i?  not  equal  to  725 
any  tenth  times,  and  hence  we  write  0  in  the  tenth's  place  of 
the  quotient;  we  then  annex  another  0  and  thereby  reduce  the 
30  tenths  to  300  hundredths,  which  we  see  is  not  equal  to  725 
any  hundredths  times,  and  hence  we  write  0  in  the  hundredths 
place  of  the  quotient ;  we  then  annex  another  0  and  thereby 
reduce  the  300  hundredths  to  3000  thousandths,  which  we  see 


136      Arithmetical  Exercises  and  Examples. 

is  equal  to  725  4  times,  with  a  remainder.  We  have  now 
obtained  the  first  significant  figure  of  the  decimal,  and  we  con- 
tinue the  division  in  the  usual  manner  to  the  sixth  decimal 
place  and  annex  the  +  sign  to  indicate  that  there  is  still  a 
remainder. 

4.     Reduce  6}  to  a  decimal. 

FIRST   OPERATION.  SECOND    OPERATION. 

6f =^  and  V=±)27.00  61=6  and  f  ;  and 

1=4)3.00 


6.75  Ans. 


.75+6=6.75  Ans. 

Reduce  the  following  fractions  to  equivalent  decimals  not 
exceeding  6  places. 

5.       |f     Ans.  .71875          9.     f  Ans.  .625 


6.  £££        Ans.  .336 

7.  Ans.  .032 


8.  f  of  |  Ans.  .107142+ 


10.  TV        Ans.  .076923+ 

11.  .37  jV     Ans.  .370625 

12.  47.18J    Ans.  47.1875 


Reduce  f  to  a  complex  decimal  of  3  places. 

OPERATION. 

3)2.000 


.6661  Ans. 

13.  Reduce  ^  to  a  complex  decimal  of  4  places. 

Ans.  .4285f 

14.  Reduce  %  to  a  complex  decimal  of  6  places. 

Ans.  .222222f 

168.  ADDITION  OF  DECIMALS. 

Since  decimals  increase  from  right  to  left,  and  decrease 
from  left  to  right  in  a  tenfold  ratio  as  do  simple  whole  num- 
bers, they  may  be  added,  subtracted,  multiplied  and  divided 
ia  the  same  manner. 

1.     Add  .785,  .93,  166.8,  72.5487  and  4.17. 


Subtraction  of  Decimals.  137 

OPERATION.  Explanation.     In  all  problems  of 

^oc  this  kind  we  write  the  numbers  so 

Q  that  units  of  the  same  order  will 

•"**  stand  in  the  same  column,  and  the 

166.8  decimal  poim  be  in  a  vertical  line  ; 

72.5487  then  we  add   as  in  simple  whole 

4  -[7  numbers. 

When  the  addition  is  completed 

.  we  point  off  in  the  sum,  from  the 
^45.Zoo7    Ans.  right  hand,  as  many  places  for  dec- 
imals as  equal  the  greatest  number  of  decimal  places  in  any  of 
the  numbers  added. 

Add  the  following  numbers. 

2.  3.25,  42.348,  748.4  and  29.32.       Ans.  823.318. 

3.  .0049,  47.0426,  37.041  and  360.0039. 

Ans.  444.0924. 

4.  1121.6116,  61.87,  46.67,  165.13  and  676.167895. 

Ans.  2071.449495. 

5.  .8,  .09,  34.275,  562.0785  and  1.01. 

Ans.  598.2535. 

6.  81.61356,  6716.31,  413.1678956,  35.14671,  3.1671 
and  314.6.  .  Ans.  7564.0052656. 

7.  l.Olf,  240.06J,  999.9,  80.6051  and  .17. 

Ans.  132^7576. 

8.  What  is  the  sum  of  the  following  numbers :  twenty- 
five,  and  seven  millionths ;  one  hundred  forty-five,  and  six 
hundred  forty-three  thousandths;  one  hundred  seventy -five, 
and  eighty-nine  hundredths  ;  seventeen,  and  three  hundred 
forty-eight  hundred-thousandths.         Ans.   363.536487. 

9.  A  farmer  has  sold  at  one  time  3  tons  and  75  hun- 
dredths of  a  ton  of  hay,  at  another  time  11   tons  and  7 
tenths  of  a  ton,  and  at  a  third  time  16  tons  and  125  thou- 
sandths of  a  ton.     How  much  has  he  sold  in  all? 

Ans.  31.575. 

169  SUBTRACTION  OF  DECIMALS. 

1.     From  345.3046  subtract  92.1435847. 


138        Arithmetical  Exercises  and  Examples. 

OPERATION.  Explanation.     In  all  problems  of  this 

345.3046  kin,d    we    write    the    numbers    so    that 

92  1435847  units   of  the  same    order  will  stand  in 

*'"  the    same    column,     and    the     decimal 

o.o   unniKQ     A  P°intS     bC    ^    *    Vertical     Hne5     theQ     WC 

^Oo.lDlUlDo  Ans.  subtract  as  in  simple  whole  numbers, 
and  point  off  in  the  difference,  from  the 

right  hand,  as  many  places  for  decimals  as  equal  the  greatest 
number  of  decimal  places  in  either  the  minuend  or  subtra- 
hend. 

Wnen  the  decimal  places  in  the  subtrahend  exceed  those  in 
the  minuend,  naughts  are  understood  to  occupy  the  vacant 
places,  and  may  be  filled  in  if  it  is  desired. 

EXAMPLES. 

81.04089        121.25       532.8  *' 
14.587         109.054:;-      9.00451681 


66.45389  Ans.     12.19562  Ans.  523.79548319  Ans. 

5.  From  461.072  take  427.125.      Ans.  33.947. 

6.  From  17.5   take   4.19.       Ans.  13.31. 

7.  From  4000.0004  take  4.3.  Ans.  3995  7004. 

8.  From  three  million  take  three  inillionths. 

Ans  2999999.999997. 
9*     From  11  take  1  and  9  thousand  trillionths. 

*  Ans.  9.999999999991. 

10.  From  24000  subtract  2.078.        Ans.  23997.922. 

11.  From  886.333  subtract  98.5427.  Ans.  787.7903. 

170.        MULTIPLICATION  OF  DECIMALS. 

1.     Multiply  26.58  by  4.3. 

OPERATION  Explanation.     In  all  problems  of  this 

yp  ro  kind  we  multiply  as  in  whole  numbers, 

and  point  off  on  the  right  of  the  product 

as  many  places  for  decimals  as  there  are 

decimal  places  in  both  the  multiplicand 

7974  and    multiplier.     The    reason    for    thus 

1063:4  pointing  off  the  3  decimal  places  in  this 

problem  is  obvious  from  the  fact  that  in 

the    multiplicand     we    have    2    decimal 

114.294   Ans.       places  or  hundredths,  which  we  used  as 


Multiplication  of  Decimals. 


139 


whole  numbers  and  thereby  produced  a  product  100  times  too. 
great;  and  in  the  multiplier  we  have  1  decimal  place  or  tenths 
which  we  also  used  as  a  whole  number  and  thereby  produced 
a  product  10  times  too  great  ;  and  both  together  gives  a  pro- 
duct 1000  times  too  great ;  hence  to  obtain  the  correct  product 
we  divide  or  point  off  3  decimal  places. 

Explanation.  In  this  operation  we 
reduce  the  factors  to  common  frac- 
tions and  then  multiplying  them 
together,  we  obtain  a  product  of 
HMf1  which  written  decimally  is 
114.294  This  process  shows  in 


SECOND    OPERATION. 

100|2658 
101  43 


114294 


1000 
or  decimally  written 

114.294  Ans. 
2.     Multiply  4.024  by  .0056. 


Ans.  another  way  why  we  point  off  on 
the  right  of  the  product  as  many 
places  for  decimals  as  there  are  dec- 
imal places  in  both  factors. 


OPERATION 
4.024 

.0056 


24144 
20120 

.0225344  Ans. 


Explanation.  In  all  problems  of  this 
kind  where  the  number  of  figures  in  the 
product  is  not  equal  to  the  number  of 
decimal  places  in  the  two  factors,  we 
must  prefix  a  sufficient  number  of  O's  to 
supply  the  deficiency.  In  this  example, 
we  prefix  one  0.  The  reason  of  this 
will  appear  evident  by  working  the  ex- 
ample as  a  common  fraction  as  shown 
in  the  second  operation  of  the  first 
problem. 

EXAMPLES. 


3.  Multiply  27  by  .9. 

4.  Multiply  .38  by  8. 

5.  Multiply  .75  by  .42. 

6.  Multiply  .OOr.  by  .0103. 

7.  Multiply  340.012  by  61.23. 
S.  Multiply  .1234  by  1234. 

Multiply  1500  l.y  .00014. 


Ans.  24.3. 
Ans.  3.04. 
\n<.  .3150. 
Ans.  .0000618. 
Ans    20818.93476. 
Aus.   152.2756. 
Ans.  .21. 


9. 

10.  What  is  the  product  of  one  thousand  and  twenty- 
five  multiplied  by   three  hundred   and    twenty  seven   ten- 
thousandths  ?  Ans.   33.5175. 

11.  What  is  the  product   of   seventy-eight   million   two 
hundred  five   thousand  and  two,  multiplied  by  fifty-three 
hundredths?  Aus.  41448651.06. 


//,  *' 
*''•>* 


140         Arithmetical  Exercises  and  Examples. 

J2.     Multiply  one  hundred  and  fifty-three  thousandths 
by  one  hundred  and  twenty-nine  millionths. 

Ans.  .000019737. 

13.  Multiply  1  thousand  by  1  thousandth.        Ans.  1. 

14.  Multiply  2  million  by  2  billionths         Ans.  .004. 

15.  What  will  37.23  tons  of  hay  cost  at  $20.75  per  ton? 

Ans.  $772.52+. 

16.  What  will    428.431    bushels  cost    at  SI.  125   per 
bushel?  Ans.  $481.98+. 

171.  To  multiply  a  decimal  or  mixed  number  by  10, 
100,  1000,  etc. 

1.  Multiply  428.375  by  100. 

OPERATION  Explanation  In  all  problems  where 

4-9£37  ^  *A  *^e  multiplier  i8  10,  100,  etc.,  we  sim- 

4400*.  0  Ans.  ply  remove  the  decimal  point  as  many 

places  to  the  right  as  there  are  naughts  in  the  multiplier,  annex- 

ing naughts  if  required. 

2.  Multiply  271.32  by  1000.  Ans.  271320. 

3.  Multiply  .756  by  100.  Ans.  75.6. 

4.  Multiply  .025  by  10.  Ans.  .25. 

5.  Multiply  61.052  by  10000.  Ans.  610520. 

172.  DIVISION  OF  DECIMALS. 

1.     Divide  17.094  by  8.14. 

FIRST   OPERATION.  Explanation.     In  all  problems 

8.14)17.094(2.1  Ans.      of  this  kind>  we  divide  as  in 
1  fi  2ft  whole  numbers,   and  then  point 

off  as   many  places  for    decimals 
from  the  right  of  the  quotient  as  the 
814  number  of  decimal  places  in  the  div- 

814  idend  exceeds  those  in  the  divisor, 

observing  to  -supply  any  deficiency 
by  prefixing     naughts.       In     this 

problem  the  excess  is  one,  and  we  therefore  point  off  one  deci- 
mal place   in   the   quotient.     The    reason   for   thus  pointing  is 
obvious  from  the  fact  that  in   the   dividend  we  had  3  decimals^ 
or  thousandths,  and  in  the  divisor  we  hadi  cU&imatjor  frcTTrtrs^  ' 
and  thousandths   divided  or  decreased    by  t^atts*  gives  Iwwi- 
dretifcks  as  a  quotient.     The  reason  will  also  appear  plain  if  we 
observe  that  the  dividend   is  the  product  of  the    divisor  and 


Division  of  Decimals.  141 

quotient  multiplied  together,  and  hence  we  point  off  enough 
decimal  places  in  the  quotient  to  make  the  number  in  the  two 
factors  equal  to  the  number  in  the  product  or  dividend,  accord- 
ing to  the  principles  shown  in  the  first  problem  of  multiplica- 
tion of  decimals 

SECOND   OPERATION.  Explanation.     In  this  opera- 


1000 

814 


1701)4  tion  we  reduce  the  decimals 


100 


to  common  fractions  and  then 
proceed  as  in  the  division  of 
mixed  numbers.      The  reduc- 
|  ^y1^-   Arns.          tion  of  the  dividend  and  divi- 
Decimally  written  2.1   Ans.      sor  to  common  fractions    and 
then    the    mixed    numbers    to 

improper  fractious,  is  performed  thus  :  the  dividend  17.094= 
17rVo4<y=Vo%9o4-5  the'  divisor  8.14==8^=fjf  This  method 
also  shows  the  reason  for  pointing  off  and  may  be  used  for  all 
problems  in  decimal  fractions. 

2.     Divide  7898.50  by  2.4G83. 

OPERATION. 

2.4083)7898.5600(3200.  Ans. 
7401!) 


49366 
19366 

oo 

Explanation.  Here  we  have  an  excess  of  decimals  in  the  di- 
visor, and  in  all  cases  of  this  kind  we  first  make  them  equal  by 
annexing  naughts  to  the  dividend,  and  the  quotient  will  be  in 
whole  numbers.  The  reason  for  annexing  the  naughts  will 
appear  more  obvious  by  solving  the  problem  in  the  form  of  a 
common  fraction. 

3.     Divide  7.07t51  by  (JS7. 

OPERATION  Explanation.     In  this  problem 

087)7. 07(J1  (  10.')  there  are  4  decimal  plares  in  the 

<>S7         .0103  Ans.    dividend  and  none  in  the  divisor, 

hence  nccording  to  the  forego- 

«;()/••  I  i"?  instruction  we   must  point 

~  oft  4  decimal  places  in  the  quo- 

tient, and  as  there  are  but  3  fig- 
ures in  the  quotient  we  prefix  1 

naught.     In  all  problems  of  this  kind,  O's  are  prefixed  to  supply 
:niy  ilvtieii'iiry  of  figures  that  may  orenr. 


14:!         Arithmetical  Exercises  and  Examples. 


4.     Divide  47.789  by  39.27. 

OPERATION.  Explanation. 

39.27)47.789(1.2168+   Ans.   lem   we   have 
3927 


8519 
7854 

GG50 
3927 


Iu  this  prob- 
,    remainder, 

after  dividing  the  dividend, 
of  6G5  ;  to  this  and  the  2  suc- 
cessive remainders  we  annex 
O's  and  continue  the  division 
until  we  have  produced  4  dec- 
imal places.  The  annexing 
of  O's  reduces  the  successive 
remainders  to  the  next  lower 
order  of  tenths  and  hence  all 
quotient  figures  produced  by 
annexing  O's  are  decimals. 
We  therefore,  point  off  from 
the  right  of  the  quotient  as 
many  places  for  decimals  as 
the  number  of  decimals  in  the 
dividend  exceed  those  of  the 
divisor,  plus  the  number  of 

O's  annexed.  This  is  done  in  all  division  problems  where  O's 
are  annexed,  and  a  sufficient  number  of  O's  should  be  annexed 
to  produce  4  or  6  decimal  places.  When  there  is  a  remainder 
after  the  last  division  the  plus  (-{-)  sign  should  be  annexed  to 
the  answer  to  indicate  that  the  quotient  is  incomplete. 

7.     Divide  .112233  by  12. 


27230 

235G2 

3GG80 
31416 


5. 


Divide  1.12233  by  12. 

OPERATION. 

12)1.12233 


8. 


9352+r=.09352  +  Ans. 
Divide  11.2233  by  12. 

OPERATION. 

12)11.2233 


9. 


.9352+  Ans. 
Divide  .0004869  by  396. 

OPERATION. 

396)4869(12+   Ans. 

396     =.0000012+. 

909 
792 

117 


OPERATION. 

12). 112233 


9352+=.009352+  Ans. 
,     Divide  112.233  by  12. 

OPERATION. 

12)112.233 


10. 


9.3627+  Ans. 
Divide  .0004869  by  3.96. 

OPERATION. 

3.96). 0004869(12+     Ans. 
396     =.00012 -f- 

909 
792 

117 


Division  of  Decimals. 


143 


11.     Divide  .0004869  by  .0396. 


FIRST     OPERATION. 

.0396).00048t>9(122-r  Ans. 
396  =.0122  + 

909 
792 

1170 
792 


378 


SECOND    OPERATION. 

396)4869(122-f-=.0122-f  Ans. 
396 

909 

792 

1170 
792 


378 


12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 


Divide  67.8632  by  32.8. 
Divide  983  by  6.6. 
Divide  13192.2  by  10.47. 
Divide  67.56785  by  .035. 
Divide  .00125  by  .5. 
Divide  7.482  by  .0006. 
Divide  1  by  999. 
Divide  84375  by  3.75. 
Divide  1081  by  39.56 
Divide  35.7  by  485. 


Ans.  2.069. 
Ans.  148.939+. 
Ans.  1260. 
Ans.  1930.51. 
Ans.  .0025. 
Ans.  12470. 
Ans.  .001001+. 
Ans.  22500. 
Ans.  27.3255+. 
Ans.  .0736+. 


21. 

22.  If  rice   costs  $.0775  per  pound,  how  many  pounds 
can  be  bought  for  $40.64875  ?  Ans.  524:5  pounds. 

23.  Sold   14.75  acres  of  land  for  $191.75.     What  was 
the  price  per  acre?  Ans.  $13. 

24.  Divide  four  thousand    three    hundred   twenty-two, 
and  four  thousand   five  hundred  seventy-three  ten-thous- 
andths by  eight  thousand  and  nine  thousandths. 

Ans.  .5403+. 

173.      To  divide  Decimal  Fractions  by  10,  100,  1000, 
etc. 

1.     Divide  48.76  by  10. 

OPERATION.  Kxpltnation.     In  all  problems  of  this 

4.876   Ans.  kind    we    simply    remove    the    decimal 

point  as  many  places  to  the  left  as  there  are  O's  in  the  divisor. 
The  reason  for  this  was  fully  shown  on  page  133.  When  there 
are  not  a  sufficient  number  of  figures  in  the  dividend  to  alkw 
this  to  be  done  naughts  must  be  prefixed  to  supply  the  defi- 
ciency. 


IN         Arithmetical  Exercises  and  Examples. 

2.  Divide  875.25  by  100.  Ans.  8.7525. 

3.  Divide  .52:n  by  1000.  Ans.  .0005231. 

4.  Divide  72  by  LOOOO.  Aris.  .0072. 

5.  Divide  9.85  by  loo.  Ans.  .0985. 
0.  Divide  .025  by  200.  Ans.  .000125. 
7.  Divide  412.99  by  10.  Ans.  41.299. 

174.     MISCELLANEOUS  PRACTICAL  PROBLEMS. 

Much  a*  occur  /;/  fhr  c<;itnfin(/  rut  nn.   fnctnry.  irurkxhop,  on 
the  plantation^  am/  in  the  nir/W.s  departments  of  bu»- 
i  life. 


1.  What  is  the  cost  of  1465  pounds  of  corn  at  84  cents 
per  bushel,  and  how  maov  bushels  are  there? 

Ans.  S21.97}  cost. 
2t;  Bush.,  9  flbs. 

2.  Sold  51*94  pounds  of  hay  at  $23.75  per  ton.     How 
many  tons  were  there,  and  what  was  the  value  of  it? 

Ans.  2  tons,  1294  Ibs. 
$62.86|  value. 

•I.     Pxui-ht  320-42  bushels  of  wheat  at  $1.95  per  bushel. 
What  was  the  cost?  Ans.  $625.  HOA. 

4.  Bought   11361   pounds  of  dried    peaches  at  $5.80 
per  bushel.     How  many  bushels  were  there  and  what  did 
they  cost?  Ans.  34  bushels.  14}  Ibs. 

$199.74ff'cost. 

5.  Bought  15  bushels  and  31  pounds  of  corn  at  78} 
cents  per  bushel.     What  was  the  cost  ? 

Ans.   $12.20if|. 

6.  Bought  3  coops  of  chickens  containing  2  dozen  and 
7  chickens  each,   at  $4.35  per  dozen.     What   did    they 
cost?  Ans.  $33.71}. 

7.  What  will  74  pounds  and  11  ounces  of  butter  cost 
at  42}  cents  per  pound?  Ans.  $31.74-^. 

8.  Bought  36  pounds  and  7  ounces  of  tea  at  $1.12} 
per  pound.     What  did  it  cost  ?  Ans.  $40.99^-. 

9.  Butter  is  worth  45  cents  per  pound.     How  much 
can  be  bought  for  20  cents  ?  Ans.  7-^  ounces. 


Miscellaneous  Practical  Problems.  145 

10.  What  is  the  cost  of  31845  feet  of  lumber  at  §22.25 
per  M.  ?  Ans.  $708.55J. 

11.  What  will  183  feet  of  lumber  cost  at  $25.75  per 
M.  ?  Ans.  $4.71TV 

12.  Bought  3  bales  of  hay  weighing  as  follows:  (1) 
421  pounds   (2)  394  pounds,  (3)  487   pounds,   at  $22.50 
per  ton.     What  did  it  cost  ?  Ans.  $14.64f. 

13.  Sold  3]   dozen  boxes  Spencerian  pens  at  $108  per 
gross.      What  did  they  amount  to?  Aus.  $29. 2.1. 

14.  How   much  coffee   can   I  buy  fur   5   cents   when  a 
pound  costs  28  cents?  Ans.  2^  ounces. 

15.  What  is  the  cost  of  400  T.  2  cwt.  3  qrs.  20  Ibs.  of 
iron  at  $60  per  ton  of  2240  pounds  ? 

Ans.  $24008.784. 

1  G.  A  planter  shipped  6  dozen  dozen  boxes  of  peaches 
to  market,  but  being  delayed  on  the  way  -I  a  dozen  dozen 
boxes  spoiled  ;  the  remainder  were  sold  it  70  cents  per  box. 
What  did  they  amount  to?  Ans.  $554.40. 

17.  Bought    12   dozen  and  5   huts  at   $11   per   dozen. 
What  did  they  cost  ?  Ans.  $136.58$. 

18.  Wh;it.  is  the  amount  due  for  the  freight  of  40000 
% pounds  of  merchandise  fur  JMJ5  miles  at  fn'  for  100  pounds 

for  100  miles?  Ans.  $193. 

11).  What  is  the  cost  of  2381 J  pounds  of  cotton  at 
17||  cents  IHT  pound  ?  Ans.  $  t24.24f|. 

20.  What   is  the   cost   of  a    14     carat   gold  chain    that 
weighs  4  o/.  7  pwt.  15  gr.  at  $1.20  per  pwt.  for  pure  gold, 
allowing  •>  <'  per  irrain  on  full  weight  for  manufacturing  and 
the  alloy?  Ans.   ^71.85*. 

21.  A    hardware  merchant  received   from   Liverpool  an 
invoice  of  iron    weighing    2   T.    2  cwt.  3  <jrs.  20  Ibs.,  long 
tori  weight;  the  invoice  price  was  £12,  17s.  Gd.  per  ton; 
What  did  the  whole  cost  in  sterling  money? 

Ans.  £27,  12s.  8£fd. 

22.  Bought  4(192  pounds  of  barley  at  $.88  per  bushel. 
How  many  bushels  were  there  and  what  was  the  cost? 

Ans.   97  bush.  M  Ibs. 
,  Cosi  £S<;.02. 


146        Arithmetical  Exercises  and  Examples. 

23.  Bought  2765  pounds  of  oats  at  76/  per  bushel. 
What  was  the  cost,  and  how  many  bushels  were  there  ? 

Ans.  $65.661  cost. 
86  bush.  13  Ibs. 

24.  What  is  the  cost  of  4878  pounds  of  wheat  at  $2.45 
percental?  Aus.  $119.511. 

25.  What  is  the  cost  of  200  sacks  of  guano  each  weigh- 
ing 162  pounds,  at  $52 i  per  ton  ?  Ans.  $846.45. 

26.  What  is  the  value  of  5790  hoop-poles  at  $18  per  M  ? 

Ans.  8104.22. 

27.  What  is  the  value  of  8750  shingles  at  $8.75  per  M? 

Ans.  $76.f>r>i. 

28.  What*  is  the  value  of   11428  fence  pickets  at  $9 
per  M?  Ans.  $102.852. 

29.  What   is  the  value   of    1364    pine   apples   at 
per  C.?'  Aus. 

30.  What  is  the  cost  of  2417   cocoanuts  at 
C.?  Ans. 

31.  What  is  the  value  of  78420   railroad  ties  at  $75 
per  M.?  Ans.  $5881.50. 

',\'2.  What  is  the  freight  on  540  bales  cotton,  weighing 
243084  pounds,  |d.  per  pound  from  New  Orleans  to  Liv- 
erpool? Ans.  £633  Os.  7 3d. 

33.  What  is  the  freight   in    United   States  currency  on 
25000  bushels  corn  from  New  Orleans  to  Liverpool,  at  24s. 
per   imperial  quarter  of  480    pounds  ;  allowing  £1   to  be 
equal  to  $4.87  ?  Ans.  $17045. 

34.  How%  many  square  feet  in  a  pavement  120  feet  4 
inches  long  and  10  feet  wide?         Ans.   12U3J  sq.  feet. 

35.  How  many  square  yards  in  a  plat  of  ground   140 
feet  3  inches  long  and  64  ieet  6  inches  wide  ? 

Ans.  1005i  sq.  yards. 

36.  How  many  squares  in  the  roof  of  a  building  78  feet 
6  inches  long,  and  48  feet  4  inches  wide  ? 

Ans.  37.94J  squares. 

37.  How  many  square   yards  of  plastering  in  the  walls 
and  ceiling  of  a  room  which  is  40   feet   6   inches  long,  24 
feet  8  inches  wide,  and  14  feet  high,  deducting  3£  square 


Miscellaneous  Practical  Problems.  147 

yards  for  doors,  windows  and  base-hoard,  and  what  will  it 
cost  at  35  cents  per  square  yard  ? 

Ans.  279ff  sq.  yards. 
$97.90ff  cost. 

38.  How  many  sq.  feel  in  8  boards,  each  measuring  16 
feet  long  and    17   inches  wide,  and  what  will  they  cost  at 
2i/ per  foot?  Ans.  181i  feet. 

$4.53J  cost. 

39.  How  many  square  feet  in  13  pieces  of  plank,  each 
measuring  -0  feet    6  inches  Jong,   14  inches  w^de  and  3 
inches  thick,  and  what  is  the  cost  at  $23  per  M.? 

Ans.  932|  feet. 
$21-453}  cost. 

40.  How  many  square  feet  in  a  circle,  the  diameter  of 
which  is  12  yards?  Ans.   1017.8784  sq.  feet. 

41.  How   many   shingles   will  it  require  to    shingle    a 
building,  the  roof  of  which  measures  44  feet  7  inches  from 
cave  to  cave,  without  allowances,  by  50  feet  4  inches  long, 
allowing  a  shingle   to  cover  a  space   4  inches  wide  and  5 
inches  long  ?  Ans.   16157. 

42.  A.  yard  is  24  feet  3  inches  long  by  11   feet  5  inches 
wide ;  how  many  brick,  4  by  8  inches  will  it  take  to  pave 
it,  no  allowance  to   be   made  for  the  openings  between  the 
bricks?  Ans.  1245$  J, 

43.  How  many  square  yards  of  paving  in  a  sidewalk  64 
feet  long  and  11  feet  8  inches  wide? 

Ans.   82|^  square  yards. 

44.  How  many  flags,    each   16    inches    square,   will   it 
require  to  flag  a  walk  22  yards  1  foot  4  inches  long  and  6 
feet  8  inches  wide  ?  Ans.   252"J. 

45.  How   many  yards   of   carpeting  that  is  27   inches 
wide,  will  it  take   to  cover   the  floor  of  a  room  that  is  25 
feet  6  inches  long,   and  22  feet  9  inches  wide,  making  no 
allowance  for  waste  in  matching  or  turning  under? 

Ans.  85j-£  yards. 

46.  A  water  pipe  is  50  feet  9  inches  long,  and  its  diam- 
eter is  30  inches  ;  what  is  its  concave  surface  ? 

Ans.  57397.032  inches. 


148        Arithmetical  Exercises  and  Examples. 

47.  How  many  cubic  feet  in  a  box  5  feet  long,  3  feet 
wide  and  4  feet  deep  ?  Ans.  60  cubic  feet. 

48.  What  is   the  freight  on  a  box  6  feet  4  inches  long, 
4  feet  wide  and  3  feet  9  inches   deep  at  25  cents  per  cubic 
foot?  Ans.  $232. 

49.  What  will  be  the  freight  on  a  box    9    feet    3   indies 
Ion-,  4  feet  6    inches   wide,    2  feet    10    inelh-    de  p.  at  30 
c« -iits  a  cubic  foot?  Ar.s.   $3r>  : 

50.  How    many  bushels   will  a  bin    hold,    that  is  10  feet 
long,  8  feet  6  inches  wide,  and  5  feet  2  inches  '.cep? 

Ans.  :;:>LVJO-|-  bushels. 

51.  How  many  cords  of  wood  in  two  ranks,  each  4-1  fret 
long  and  6  feet  3  inches  high?  Ans.    1  7  ,:l(.  cords. 

52.  How  many  barrels  will  a   (jiiadi  ilatcral   cistern  hold, 
whosejieight  is  12  feet  and  width  of  side  f>  icet  S  inches? 

Ans.    HI-,*;2,1;  ban 

53.  How  many   cubic   yards  in  a  levee  80  rods  loiiir,  «IO 
feet  wide  at  the  base,   12-j  feet  at  the  top,  and  5  feet    1  ineln  s 
average  depth?  Ans.   IM.V 

54.  How    many  gallons   will  a  box   hold,   that  is  f>  fed 
long.  2  feet  4  inches  wide,  and  3  feet  deep  ? 

Ans.  201.81  +   gallons. 

55.  How  many  cubic  feet  in  a  cylinder  U  fret  long  3  ieet 
4  inches  in  diameter?  Ans    52.36  cubic  feet. 

50.      IIow  many   gallons  in  a  cylindrical    cistern,  9  feet  b* 
inches  high,  and  7  feet  2  inches  in  diameter? 

Ans.   28(Hi.<)S!Mi  gals. 

57.  How  many  pints  in  a  cylindrical  vessel,  whose  height 
is  14  inches  and  diameter  12-1  inches? 

Ans.  59.5  pints. 

58.  How  many  bushels  in  a  cylinder  shaped  box.  whose 
height  is  10  feH,  and  diameter  10  feet? 

Ans.  031.125  bu. 

59.  How  many  cubic  feet  in  a  frustum  of  a  cone,  whose 
height  is  6  feet,  diameter  of  the  greater  end  is  4  feet  and  of 
the  smaller  end  3  feet?  Ans.   58.1196  cubic  feet. 

60.  How  many  gallons  in  a  cistern  which  is  in  the  form 
of  a  frustum   of  a  cone,  whose   height  is  9  feet  0  inches, 


Bills  and  Invoices.  149 

lower  base  7  feet  2  inches,  and  upper  base  6  feet  8  inches  ? 

Ans.  2671.3392  gals. 

61.  A  farmer  has  a  heap  of  grain  in  a  conical  form,  the 
diameter  of  which  is  14  feet  4  inches,  and  the  depth  5  feet 
3  inches ;  how  many  bushels  does  it  contain  ? 

Ans.  226.906. 

62.  A  barrel  i*  26  inches  long,  17  inches  in  diameter  at 
the  head,  and  20  inches  in  diameter  at  the  bung  or  center. 
The  staves  have  a  medium  curve.     How  many  gallons  will 
it  hold?  Ans.  3 1. 244 -f- gallons. 

For  full  in  ormation  and  a  thorough  elucidation  of  all 
questions  pertaininir  to  the  mensuration  of  surfaced  and 
solids,  as  contained  in  the  foregoing  miscellaneous  problems, 
and  in  the  following  bills,  see  Soule's  Philosophic  Commer- 
cial and  Exchange  Calculator,  pages  741  to  796. 

174.  BILLS  AND  INVOICES. 

Bills  in  a  general  sense,  embrace  all  written  statements 
of  accounts  and  many  legal  instruments  of  writing ;  but 
in  a  more  common  and  limited  sense  they  are  statements  of 
goods  sold  or  delivered,  services  rendered  or  work  done,  with 
the  price  or  value,  quality  or  grade  of  each  article  or  item. 
They  should  state  the  place  and  date  of  each  sale,  the  names 
of  the  buyer  and  seller,  the  extra  charges  or  discount  to  be 
allowed,  and  the  terms  of  the  sale. 

When  goods  are  bought  to  sell  again,  or  when  bills  are 
rendered  to  a  jobber  or  retailer,  or  consigned  to  an  agent, 
the  bill  is  then  called  an  invoice. 

It  is  the  custom  of  accountants  and  merchants,  when  making 
bills  to  commence  the  name  of  each  article  with  a  capital. 

When  a  charge  is  made  for  the  box,  barrel,  jar  etc.  contain- 
iQg  goods,  it  is  customary  to  write  its  price  above  and  to  the 
right  of  it  and  add  the  same  to  the  cost  of  the  goods  it  con- 
tains. 

In  making  extensions,  fractions  of  cents  are  not  used  in    the 


150        Arithmetical  Exercises  and  Examples. 

product  ;  when  they  are  \  or  more  they  are  counted  cents 
when  they  are  less  than  J  they  are  not  counted. 

In  making  the  following  bills,  students  should  use  pen  and 
ink  and  give  earnest  attention  to  the  proper  form  and  spacing, 
to  plain,  neat  and  rapid  penmanship  of  both  words  and  figures, 
and  above  all  to  the  accuracy  of  extentions  and  additions. 

When  notes  or  bills  of  exchange  are  given  in  payment,  the 
student  should  draw  the  same  and  correctly  mature  them. 

No.    1. 
NEW  ORLEANS,  Jan.  2,  1877. 


H.  A,  &  R.  C.  Spencer, 


TERMS— Cash. 


Hot.  of  A.  L  &  E.  Soule. 


1876 

Dec. 

u; 

2  bags  Rio  Coffee,  325  Ibs.  @ 

$  23ic. 

$  76 

38 

50  c. 

1  bbl.  Sugar,  234J  Ibs. 

9  c. 

21 

61 

i  Chest  Black  Tea,  35  tbs.    " 

87  Jc. 

30 

63 

1  bbl.  Rice,  24^-1  6  =227" 

8  c. 

18 

16 

40  gal.  N.  0.  Molasses 

75  c. 

30 

00 

6  doz.  Brooms 

4.15 

24 

DO 

3  bbls.  XXX  Family  Flour" 

8.12J 

24 

38 

25  Ibs.  Cream  Crackers 

16  c. 

4 

00 

50  Ibs.  Graham  do                 " 

15  c. 

7 

50 

20  Ibs.  W.  Butter 

30  c. 

6 

00 

Rec'd  pay't, 


$243.56 


A.  L.  &  E.  SOULE, 
Per  S.  Richardson. 


Bilk  and  Invoiees. 


No.  2. 


151 


Nfcw  ORLEANS,  Jariy  31,  1877. 

S.  S.  Packard  and  E,  G,  Folsom, 

Bot.  of  W.  E.  and  Frank  Soule. 

TERMS— Note  at  30  days. 


1877 

Jan. 

31 

453  £  Ibs.  Mocha  Coffee,     @ 

$  25  c. 

241           Rio  Coffee, 

18|c. 

31  6  1         C.  Sugar, 

12Jc. 

72           Duryea's  Starch, 

6ic. 

64           N.  Y.  C.  Cheese, 

17ic. 

52           W.  F.  Cheese, 

15  c. 

180           B.  Sugar, 

7*c. 

80  doz.  C.  Eggs, 

37  ic. 

42  gals.  N.  0.  Molasses, 

62£c. 

320  Ibs.  G.  Butter, 

35  c. 

23    "    Almonds, 

27  c. 

76    «    Y  H.  Tea, 

74  c. 

68  boxes  Shrimp, 

48  c. 

84  boxes  Lobsters, 

34  c. 

92  gals.  N.  0.  G.  Syrup, 

96  c. 

114    «     B.  Whiskey, 

1.08 

112  bags  Salt, 

93  c. 

320  Bbls.  Sweet  Potatoes, 

1.25 

S2  kits,  No.  1  Mackerel, 

i50 

63    Ibs.  S.  Crackers, 

11  c. 

24f    "    P.  L.  Soap, 

8Jc. 

18i    "    Codfish, 

9Jc. 

Drayage  $31.25,  boxes  $2.50. 

Rec'd  pay't  by  note  at  60  days,    $1,492  09 
W.  H.  &  F.  SOULE. 

per  J.  J.  Manson. 


NOTE.— All  of  the  extensions  of  this  bill  should  be  made  mentally 
•apid  mental  work  see  SoulS's  Contractions  In  Numbers. 


For 


Arithmetical  Exercises  and  Examples. 

No.  3. 
NEW  ORLEANS,  Jariy  31,  1877. 

Bot.  of  J.  B.  Cundiff. 


J.  M.  Butchee, 

TERM  r.Mlit. 


875  bbls.  Nes.  Potatoes  @ 

$  4.25 

H()    l    P.  B.  Potatoes,     » 

3.87  J 

325    l    Perfect'  n  Flour, 

8.50 

i:*.24    •    St.  L.  XX    " 

6.62J 

1  1  2    l    F'ily  Clear  Pork   ' 
650    '    Prime  Pork, 

17.50 
13.75 

220  kegs  Pig  Feet, 
1LM  h.lfbblsF.M.  Beef,   l 

7.50 
11. 

1ST  2  Ibs.  Choice  Ham, 
289    "    B.  Bacon, 
10(J  Piir  Tonnes, 

14  c. 
9Jc. 
8  c. 

Rec'd  pay't,        $31167  27 

NOTE— All  the  extensions  of  this  bill  shall  be  made  mentally. 


E.  C.  Spencer  &  Co., 

TERMS— Cash. 


o.  4. 
NEW  YORK,  Dec.  8,  1876. 

Bot.  of  B.  D.  Rowlee  &  Co. 


20  doz.  Missionary  Bibles,  @  $15.25 

108    "  small  New  Testam't,  u       2.50 

65    "  Prayer  Books,  ><       2.25 

65    "  Hymn  Books,  "       3. 

3  Bible  Dictionaries,  "       4. 

i  doz.  Webster's  Dict'ry     u     50. 


Rec'd  pay't,         $  953  25 

B.  D.  ROWLEE  &  CO. 

Per  E.  Conrad. 


Bills  and  Invoices. 


153 


Wm.  Melchert  &  Co., 


No.  5. 
NEW  ORLEANS,  Jariy  31,  1877. 


Bot.  of  L  L  Willi&ms  &  Go. 


321 

Ibs. 

Tobacco  Low  Lugs, 

(a).    6  c. 

1140 

u 

" 

Med.  Lugs, 

-     7Jc. 

509 

u 

i 

Low  Leaf, 

'     9ic. 

965 

u 

• 

Med.  Leaf, 

1   11  Jc. 

398 

u 

i 

Good  Leaf, 

'    13fc 

2416 

u 

i 

Fine  Leaf, 

<   15  c 

713 

U 

t 

Selections. 

'   16|c. 

Bryant  and  Nelson, 

TERMS— Dft.  30  days. 


Rec'd  pay't,    •    $ 

L.  L.  WILLIAMS  &  CO. 
No.  6. 
NEW  ORLEANS,  Dec.  17,  187 1). 

Bot.  of  Wm.  Horn  &  Go. 


1876 

1420  Ibs.  Su^ar,  Common,      @    5£c. 
1927     '        "     Good,                  l    71c. 
2810    4        ^     Fair.                    '     7ic. 
902     <         "'Prime,                 '     8Jc. 
813     '         "     Choice,                «     9}c. 
2741     '         «     Yellow  Cent'al   '  lOJc. 

Rec'd  pay't,          $ 

No.  7. 

NEW  ORLEANS,  Dec.  23,  1876. 
Montgomery  &  Lettellier, 
TERMs-3  mos.  Bot.  of  Sadler  &  Smith. 


1  Gross  Chewing  Tobacco  @  $13. 
180  Ibs.  Smoking  do  1.40 

6  M.  Havana  Cigars  •'  70. 

2  M.  N.  0.  Manufacture  do  :>  30. 

Rec  d  pay?t. 


154         Arithmetical  Exercises  and  Examples. 


No.  8. 

NEW  ORLEANS,  Jariy  21,  1877. 
Jones  and  Carpenter/ 

Bot.  of  Stewart  &  Henderson. 

TERMS— Note  60  days. 


1877 


34  bbls.  La  Orange,  large,  @  $5.75 
27  boxes  Messina  Lemons,  "  6.00 
03  cases  Malaga  Grapes,  "  1.75 
45  boxes  California  Pears,  "  4.50 
5  mats  Dates,  593  Ibs.,  "  7ic. 


Rec'd  pay't,  by  note  at  60  days.       $ 

STEWART  &  HENDERSON. 

No.   9. 
NEW  ORLEANS,  Nov.  17,  1876. 


Geo.  B.  Brackett  &  Co., 


Bot.  of  R.  Spencer  Soule, 


TEKM8— Cash. 


1427  bu.No.l  Winter  Wheat®  $1.55 

856  "    No.  2  Winter  Wheat  "     1.47 

420  "    111.  No.  1  White  do  "     1.41 

3145  "    W.  Corn  "       .70 

1040  a    B.  Oats  "       .55 


Rec'd  pay't,  $ 

R.  SPENCER  SOULE. 

No.   10. 
NEW  ORLEANS,  Feb.  J.,  1877. 

F.  L  &  W.  P.  Richardson, 

Bot.  of  P.  W.  Sherwood  &  Co. 

TERMS— 1  mo . 


;0  box.  Sperm  Candles,  596  Ibs.  @  .35J 
24  do.  Adam.  Ex.  do.  483    "     "  .28 
15  do.  Sil.  Gloss  St'rch  360    «     «  .lOfi 
Rec'd  pay't.  : 


Bills  and  Invoices. 


155 


Eeald  &  Howe, 

TERMS— Due  Bill  1  mo. 


No.  11. 

NEW  ORLEANS,  Jan.  19,  1877. 

Bot.  of  Cole  &  Montague. 


342  Ibs.  La.  Pecans                   @ 

$  .13 

289  "  Taragona  Almonds 
175  "  Naples  Walnuts              " 
196  "  Brazil  Nuts                     " 

.21 
.17 
.11 

268  "   Western  Chestnuts 

.18 

160  Boxes  Figs 
585  Cocoanuts  @  $45  per  M. 
61  Bunches  Bananas 

.20 
1.75 

14       do.       Plantains               " 

.85 

327  Pine  Apples  @  $80  per  M. 

Hibbard  &  Gray, 

TERMS— Cash. 


Rcc'd  pay't,  $ 

COLE  &  MONTAGUE. 


No.    12. 
NEW  ORLEANS,  Feb.  1,  1877. 

Bot.  of  Odell  &  Faddis. 


2714  Ibs.  Black  Moss                @    4  / 
1829    "    Gray  do.                       «     1J/ 
913    "    Wool                           <•  24  / 
74    "    Live  Geese  Feathers  "  65  / 
1528  Packages  Broom  Corn      "     6  ^ 
752  Ibs.  Baling  Twine              "  14  / 
800  yds.  Indian  Bagging          "  11  ft 

Rec'd  pay't,  $ 

ODELL  &  FADDIS, 

Per  C.  P.  Meads. 


156        Arithmetical  Exercises  and  Examples. 


No.  13. 

NEW  ORLEANS,  Jan.  7,  1877. 
Spaulding  &  Musselman, 

Bot.  of  Warr  &  Bogardus. 

TERMS— 60  days. 


78i  yds.  Black  Silk,         @  $2.90 

148         Muslin,                    <         16  c. 

62         Cassimeres,                    1.75 

-38}       Blk.  F.  Broadcloth        5.25 

45           "      "  Doeskin,           1.50 

324         American  Satinets,   '       95  c 

3  cases,  each  40  psr  Amos- 

keag  Sheetings, 

11231    1 

1204    L  3423}  yds.       "        IT-'.c. 

10962   j                      J 

142  yds.  6-4  Alpaca,            "       32  c. 

560     "  Union  Ginghams,    "       ll}c. 

491      "  Am.  Fancy  Prints,  "       12  c. 

107     "  Manch'ter  Delains,  "       21}c. 

. 

10  doz.  Handkerchiefs,       "     2.15 

Ladies  Hose.                  * 

2  ps.  61  j  yds.  Can.  Flannel  u      18  c. 

Rec'd  pay't,         \ 

\ 

Tasker  &  Felton, 

TERMS— Cash. 


Bills  and  Invoices.  157 

No.  14. 
NEW  ORLEANS,  Feb.  8,  1877. 

Sot.  of  Allen  &  Shields. 


50 

Ibs.  Casing  Nails, 

@ 

$      7c. 

i; 

duz.  Mortice  Locks, 

u 

7.50 

* 

"     Porcelain  Knobs, 

U 

4.75 

5(1 

pr.  Butts, 

U 

25c. 

•' 

Gross  Screws, 

u 

75c. 

I   8 

bars,  \\  X  2  Bar  Iron, 

254  Ibs., 

u 

5c. 

2 

Rowland  No.  2  Spades, 

u 

1. 

Lee  and  Ward, 

TERMS— Due  end  mo. 


Rec'd  pay't,         $ 

ALLEN  &  SHIELDS. 

No.   15. 
NEW  ORLEANS,  Mar.  9,  1877. 

Bot.  of  Harris  &  DeRussy, 


?>  reams  Cap  Paper,  -@  $3.25 

2  doz.  Ebony  Rulers,  3.50 
4  6  qr.  Med.  Ledgers  24  qrs.          1.75 
33"  Demy  Journals,  9   "  1.25 
33  "    «    Cash  Books,  9  "  1.25 

3  6  "    "    Sales  Books  18"  1.15 

4  gross  Pen-holders,  2.10 

1  doz.  Black  Ink,  4.50 
•}  ream  Blotting  Paper,  2.5() 

2  doz.  Mucilage,  2.75 

3  "    Carmine,  2.15 

doz.  Bill-books,  4.25 

Rec'd  pay't, 


158         Arithmetical  Exercises  and  Examples. 

No.  16. 
NKW  ORLEANS,  April  1,  Jf<S77. 


E.  J.  &  R.  Paul, 

TERMS— Due  bill  30  days. 


Bot.  of  Gresham  &  Harp. 


J  doz.  Comb's  Con.  of  Man,  (a), 

H2. 

2  ."  Dana's  Geological  Story 

briefly  told, 

10. 

6       Soule's  Phi.  Arithmetics 

42. 

6           "     Con.  in  Numbers, 

if 

6           "     Prim.  Arithmetic 

9. 

}       Webster's  Acad.  Diet., 

18. 

2       Swinton's  Lan.  Lessons, 

3.25 

li  '  Steel's  Nat.  Philosophy, 

12. 

i    "  Spencer's  Science  of 

Sociology,                        " 

10.50 

2  copies  Wood's  Byron, 

3. 

4      k'     Dick's  Shakspeare,     '• 

4.50 

dray  age  75^,  box  50/. 

T.  Janney, 


Eec'd  pay't.       $ 

GRBSHAM  &   HARP. 

No.   17. 

NEW  ORLEANS,  Jan.  1,  1876. 


To  A.  Laborde. 


Dr. 


For   furnishing,    making,    and    laying|| 
Bru  sels  carpeting  22  inches  wide|| 
in  2  rooms  measuring  as  follows  :'| 
No.  1.  24  ft.  3  in.  X  20  ft.  9  in. 
No.  2.  18ft.  6  in.  X  17  ft.  8  in. 
*  — yards,  @  $1.95 

68          "     6-4  Che.  Mat,     "         90c. 
Rec'd  pay't.  $ 

*  In  this  bill  no  allowance  is  made  for  waste  in  matching  or  otherwise. 


Bills  and  Invoices. 


159 


Warner  &  Cornell, 

TERMS— Note  at  60  days. 


No.  18. 
NEW  ORLEANS,  Jan.  25,  1877. 

Bot.  of  Goldsmith  and  Clark. 


2143  'l.s. bush.  Yellow 

Corn.  @  $    61c. 

12-J1  ll.s.-  "    Texas 

Wheat.  »       1.70 

852  ibs.—  "    White 

Oats.  "        54c. 

7-!l  Ibs. u    Barley"        83c. 

1427    "  —     — Cwt.  Bran,  "        75c. 

-        —Tons  Timo 
thy  Hay,  "   18.50 

1701  Ibs.—       -Tons  Clover 

Hay,  "  20. 


Rec'd  pay't,  by  note  at  60  days.       $ 

GOLDSMITH  AND  CLARK. 

No.   19. 

NEW  ORLEANS,  Jan.  3,  1877. 
Geo.  F,  Bartley  &  Co., 

To  Steamship  Knickerbocker  and  Owners,  Dr. 


For  Freight  on cubic  feet  @  25/ 

The  same  bein^r  contents  of  8  boxes 
measuring  as  follows : 
Nos.  1,  2  &  3,  5  ft.  4  in.  x  4  ft.  6  in.  x 

2  ft.  8  in.  = 
Nos.  4,  5  &  (I,  (>  ft.  2  in.  x  3  ft,  0  in.  x 

2  it.  11  in.  = 
Nos.  7  &  8,  12  ft.  3  in.  x  2  ft.   1  in.  x 
1  ft.  6  in.  = 

'      Rec'd  pay't, 


160        Arithmetical  Exercises  and  Examples. 
0.  Emmet  &  Co., 


No.  20. 
NEW  ORLEANS,  Jan.  4,  1877. 


To  B.  Criswell. 


Dr. 


For  rent  of  house  No.  386  Dryades 
St.,  from  Oct.  7,  1876,  to  Jan.  1, 
1877,  2ff  months,  @  $35 

For  services  as  collector  from  Sept.  19, 
187*,  to  Jan.  4,  1877,  both  inclu- 
sive, 3-J-|  months,  @  $75 

Rec'd  pay't, 
No.  21. 


The  la.  Levee  Co., 


NEW  ORLEANS,  Jan.  9,  1877. 
To  James  Selleck  &  Co. 


For  Constructing  cubic  yards 

Levee  @  45/  as  per  the  follow- 
ing measurements : 
1st  Section  890T%  ft.  long,  70  ft.  wide 
at  the  base  and  30  ft.  at  the  top, 
with  an  average  depth  of  8-^  ft. 
2nd  Section  165  ft.  long,  60  and  25 
it.  respectively  for  the  lower  and 
upper  widths,  and  6,  7?,  5i,  8,  9, 
and  6*  ft.  in  depth  at  different 
points. 

For  Excavating  -  —  cubic  yards 
Earth  @  45/,  the  same  being  the 
contents  of  a  cellar  measuring  as 
follows : 

92  ft.  long  and  50  ft.  wide  at  the  top 
and  86  ft.  long  and  44  ft.  wide  at 
the  bottom,  average  depth  8  ft.  4  in 

Rec'd  pay't, 


Bills  and  Invoices. 


161 


Geo,  Soule, 


No,  22. 
NEW  ORLEANS,  Jan.  16,  1877. 


To  Clark  &  Eofeline. 


Dr. 


For  composition  Arithmetical  Exercises 
and  examples  180  pp.,  1600  ems  a 
page,  @  75c.  per  m. 

For  press  work  on  54  token,  @  $  50c. 

For  4  Reams  Paper,  u     6.00 

For  Binding  500  sep.,  "       25c. 


Rec'd  pay't,  $ 

CLARK  &  HOFELINE. 


No.  23. 

NEW  ORLEANS,  Jan.  18,  1877. 

Western  Union  Telegraph  Co., 

To  Jacob  Simon  &  Co.,  Dr. 


For 


cubic   feet  Timber 


$24  per  100  ;  the  same  being  the 
contents  of  50    Telegraph    polos 
measuring  as  follows : 
40    Poles   are   70    feet    long,     16x16 
inches  at  the  larger  end  and  so  12- 
*inain  for  a  distance  of  10   feet,  at 
which   point  they  begin  and   taper 
regularly  to  the  smaller  cud',  whi(-h 
is  6x6  inches. 

10    Poles   are    60    feet    long,    16x12 
inches  at  the  larger  end,  6x4  inches 
at  the  smaller  end,  and  ti  jer  reg-i 
larly  the  whole  length. 

Rec'd  pay't. 


162        Arithmetical  Exercises  and  Examples. 

No.  24. 

NEW  ORLEANS,  Jariy  29,  1877. 


S.  Drey  fuss, 


To  V.  Keiffer. 


Dr. 


Jan. 

1 

To  old  balance,  as  per  bill 

rendered. 

91 

10 

6 

12  cords  Ash  Wood 

@$7. 

84 

00 

6 

4  cords  Oak  Wood 

"  $6.50 

26 

00 

14 

50  bbls.  Pittsburg  Coal 

"        60c. 

30 

00 

$231 

10 

Cr. 

8 

By  Cash 

$50 

21) 

"  6  days  Labor,  at  $4, 

824 

74 

00 

Balance  due  Jan.  2(J,  1877.            $  157   10 

Settled  by  note  at  60  days. 

V.  KEIFFER. 

A.  &  S.  H.  Souk, 


No.  25. 
NEW  ORLEANS,  Jan.  12,  1877. 


To  Z,  M.  Pike  &  Co. 


Dr. 


To  4378  feet  Com.  Boards        @ 

$21.  per  m. 
"  1760  feet  Dressed  Flooring  " 

$28.50  per  m. 
"  5125  Bricks  " 

$14.25  per  m. 
"  9250  Cypress  Shingles 

$6.50  per  m. 
"  Cartage  and  Labor 

Rec'd  pay't, 


14 


25 


Levy  &  King, 


Bills  and  Invoices.  163 

No.  26. 
NEW  ORLEANS,  Dec.  81,  1876. 

To  R.  &  C.  Rice,  Dr. 


For   --   sq.    yds.     North     River 
Flags  @  $7.50  as  per  the  follow- 
ing measurements  : 
Nos.  1,  2  &  3,  are  each  4  ft.  3  in.  by 

3  ft.  6  in.  =  sq.  ft. 

Nos.  4,  5  &  6   are   each   4  ft.  8  in.  by 

3  ft.  4  in.  =  sq.  ft. 

Nos.  7  &  8  are  each  4  ft.  0  in.  by  3  ft. 

0  in.  =  sq.  ft. 

Nos.  9,  10  &  11  are  each  3  ft.  4  in.  by 

2  ft.  9  in.  =  sq.  ft. 


For 


sq.  yds.  German  Flags  @ 
$2.25,  comprising  152  Flags,  each 
22x16  inches. 

For  ---  sq.  yds.  Brick  Pavement 
@  $1.15,  contained  in  a  side- 
walk measuring  124  ft.  4  in.  long 
by  11  ft.  9  in.  wide. 
For  124  ft.  4  in.  Curbing  ty  $  1.30 
For  -  cu.  yds.  Granite  u  $16.00 
contained  in  23  blocks  of  stone 
measuring  as  follows  : 
Nos.  1  to  7  inclusive,  are  each  26xl£x 

10  inches,  = 
Nos.  8  to  20  inclusive,   are    each  2ox 

16x9},= 

Nos.    21    to    23    inclusive,    are   each 
42x35x21  inches,  = 


Rec'd  pay't,         $ 


164        Arithmetical  Exercises  and  Examples. 


No.  27. 

NEW  ORLEANS,  Jan.  28,  1877. 

Invoice  of  Sundries,  purchased  by  J.  Simmons  &  Co.  and 
shipped  per  Steamer  La  Belle,  for  acc't.  and  risk  of  James 
Byrnes,  Shreveport,  La. 


87  bbls.  Molas's,  3498  gals.  @ 

60c. 

20  hhds.  Sugar,  23780  Ibs.  " 

9c. 

10  bbls.  Rice,       2150  Ibs.  " 

5c. 

4346 

50 

Drayage 

17 

50 

Insurance  on  $4800.40      " 

l%- 

35 

00 

Commission  on  $4364.00  " 

21%. 

109 

10 

4503  10 


No.  28. 


W.  H.  Carey. 


NEW  ORLEANS,  Jan.  14,  1877. 
To  Geo.  Jumonville,  Dr. 


For  Slating  a  roof  measuring  72  ft.  4 
in.  by  49  ft.  10  in.  and  contain- 
ing   squares  (a/,  $14.50 

For  239  ft.  Guttering  .90   

Rec'd  pay't. 
No.  29. 

NEW  ORLEANS,  Feb.  1,  1877. 
Mississippi  Valley  Transportation  Co., 

To  Buck  &  Richardson,  Dr. 


For  Servises  rendered  in  cause  No. 
55472.  "Steamer  R.  E.  Lee  and 
owners  vs.  Miss.  V.  T.  Co." 

Rec'd  pay't. 


Bills  and  Invoices.  165 

No.  30. 


NEW  ORLEANS,  Nov.  4, 
New  Orleans,  'St.  Louis  and  Chicago  R.  R., 

To  W.  L.  &  E.  Sail,  Dr. 


For  150  Cisterns  holding 


@  2J^  per  gal.  The  inside 
measurement  of  each  cistern  is  as 
follows:  11  ft.  o  in.  perpend  cular 
height,  lower  base  9  ft.  '1  in.  and 
upper  base  8  ft.  5  in. 

Rec'd  pay't. 

TABLES  OF  WEIGHTS  AND  MEASURES. 

175.  Weight  is  that  property  of  bodies  by  virtue  of 
which   they  tend  toward  the  center  of  the  earth,  and  the 
resistance  required  to  overcome  this  centralizing  pressure, 
or  gravitating  tendency  of  bodies,  is  what  is  named  weight; 
Weight  varies  according  to  the  quantity  of  matter  a  body 
contains,  and  its  distance  from  the  centre  of  the  earth. 

176.  A   Measure  is  a  standard  unit  established  by 
law  or  custom,  by  which  quantity,  such  as  extent,  dimension, 
capacity,  amount  or  value,  is  i:  easured  or  estimated. 

There  are  seven  kinds  of  measure : 

1st.  Length.  2d.  Surface  or  Area.  3d.  Solidity  or 
Capacity.  4th.  Weight  or  Force  of  Gravity.  5th.  Time. 
6th.  Angles.  7th.  Money  or  Value. 

COMPARISON  OF  STANDARD  UNITS  OF  MEASURE. 

1.  The  Yard— 3  feet—  36  inches. 

2.  The  Meter— 3.2808  feet—  39.37  inches'. 

3.  The  Vanir-2.7778  feet—  33J  inches! 

4.  The  Troy  and  Apothecaries  pound=12  oz.=:5760  grains. 

5.  The  Avoirdupois  pound=16  oz.=  7000  grains. 

6.  The  Wine  gallon=  231   cubic  inches. 

7.  The  Beer  gallon— (nearly  obsolete)—     282  cubic  inches. 

8.  The  Dry  gallon—  268.8  cubic  inches. 


H5(i        Arithmetical  Exercises  and  Examples. 

9.     The  Imperial  gallon  of  England—    277.274  cubic  inches. 

10.  The  Bushel=r:4  pks.^32  qts.=64  pts.=: 

2150.42  cubic  inches. 

11.  The  Imperial  Bushel  of  England— 2218.192  cubic  inches. 

12.  The  Diamond  grain  is  equal  to  .8  of  a  grain  Troy. 

13.  The    Gallon,    wine   measure,    of  distilled    water    weighs 
8.3388  pounds  Avoirdupois  or  10.134  pounds  Troy. 

14.  The   Civil  Dan  rrmmriK •<.-.<  ami  oiids  at  midnight,  and  the 
Astronomical  Day,  at  noon  of  the  Civil  Day. 

15.  The   Kntur   /><///  is  the  interval  of  time  between  two  suc- 
cessive  p:is-:ijri's  of  the  sun  across  the    meridian  of  any  place, 
and  they  are  of  unequal  length  on  account  of  the  unequal  or- 
bital motion  of  the  earth  and  the  obliquity  of  the  ecliptic. 

RATIOS. 

When  the  diameter  of  a  circle  is  1,  the  circumference  is 

3.1416. 
Whenthe  area  of  a  square  is  1,  the  area  of  a  circle,  the  diameter 

of  which  is  equal  to  one  side  of  the  square,  is  .7854. 

When  the  solidity  of  a  cube  is  1,  the  solidity  of  a  sphere,  the 

diameter  of  which  is  equal  to  one  side  of  the  cube,  is    .5236. 

\\K1GHT  OF  COIN. 

$10000  Gold=258000  gr.=44  Ibs.  9  oz.  10  pwt.  0  gr.   Troy. 

$1000  Silver  dollars  old  issue=412500  gr.=71  Ibs.  7  oz.  7 
pwt.  12  gr. 

$1000000  Gold  weigh  53750  ounces  Troy  or  3685.71  -Avoir- 
dupois pounds. 

$1000000  Silver  Trade  dollars  weigh  875000  ounces  Troy  or 
60000  pounds  Avoirdupois. 

$1000000  Silver,  half  and  quarter  dollars,  20  cent  pieces  and 
dimes,  weigh  803750  ounces  Troy  or  55114.28  Avoirdupois 
pounds'. 

VALUE  OF  COIN. 

Gold  coin=about  86  cents  per  pennyweight. 
Silver  com=about  $1.11  per  ounce. 

For  more  extended  tables  of  weights  and  measures,  a 
condensed  history  of  time  measure  and  the  units  of  measure 
in  use  in  the  early  ages  of  civilization,  see  Soule's  Philoso- 
phic, Commercial  and  Exchange  Calculator,  pages  121  to 
152  and  497  to  519. 


Tables  of  Weights  and  Measures. 


167 


'I._UNITED  STATES  MONEY. 


10  Mills   (m.) 
10  Cents 
10  Dimes 
10  Dollars 


=  1  Cent,  f 

—  1  Dime,  d. 

=  1  Dollar,  $ 

--=  1  Eagle,  E. 


II.— ENGLISH  MONEY. 

TABLE. 

4  Farthings  (far.)  —  1   Penny,  d. 

12  Pence  =  1   Shilling,  a. 

20  Shillings  =  1  Pound  (Sovereign,)  £ 

£1  =  $4.8665 


III.— FRENCH  MONEY. 

TABLE. 

10  Centimes  = 

10  Decimes  or  100  Centimes     — 
Fr.  1 


1   Decime. 
1  Franc. 
19  3  Cents. 


TIME  MEASURE. 


60  Seconds  (a.) 
60  Minutes 
.24  Hours 
7  Days 

365  Days 

366  Days 
100  Years 


Minute,  m. 

Hour,  h. 

Day,  d. 

Week,  wk. 
Common  Year,    yr. 

Leap  Year,  yr. 

Century,  c. 


The  names  and  orders  of  the  months,  and  the  number  of  days 
contained  in  each,  are  now  as  follows; 


Names. 

No. 

No.  days.  '        Names. 

No. 

No.  days. 

January, 

1st, 

31 

July, 

7th, 

31 

February 

2d, 

28 

August, 

8th, 

31 

March, 

3d, 

31 

September, 

9th, 

30 

April; 

4th, 

30 

October, 

10th, 

31 

May, 

5th, 

31 

November, 

llth. 

30 

June, 

6th, 

30 

December, 

12th, 

M 

ir>^        Arithmetical  Exercises  and  Examples. 

The    number  of  days  in  each,  may  be  readily  remembered  by 
committing  to  memory  the  following  lines  : 

"  Thirty  days  hath  September, 
April,  June,  and  Novemlu-r ; 
And  all  the  resth;i\<   thirty-one, 
Save  F«'l.ru;iry,  \vhich  alone 
Huth  twenty-eight;  and  this,  in  fine, 
One  year  in  four  hath  twenty -nine." 


(LINE)  LINEAR  OR  LONG  MEASURE. 

TABLE. 

12  Inches  (in.)  =     1  Foot,  ft. 

3  Feet  =     1   Yard,  yd. 

5J  Yards  or  16Jifeet   =     1  Rod  or  Pole,  rd.  or  po. 

40  Rods  =     1   Furlong,  tur. 

8  Furlongs  =     1  Mile  (Statute  Mile)  m. 

3  Miles  =     1   League,  lea. 


MARINERS'  MEASURE. 

TABLE. 

6  Feet  =     1   Fathom. 

120  Fathoms  =     1  Cable-length. 

880  Fathoms  or  7J  Cable-lengths  =     1   Mile. 

A  knot,  or  geographical  mile,  is  -fa  of  a  degree,  and  is  equiva- 
lent to  1.15257  statute  miles. 

The  length  of  a  degree  at  the  Equator  is  nearly  equal  to  69J 
statute  miles.  The  length  of  an  average  degree  on  the  meridian 
is  69.042  statute  miles. 


MISCELLANEOUS  UNITS  OF  LINEAR  MEASURE. 


of  an  Inch  =  A  Line  (American). 

of  an  Inch  =  A  Line  (French). 

4  Inches  =  A  Hand. 

3  Inches  =  A  Palm. 

9  Inches  =  A  Span. 

3  Feet  =  A  Pace. 

2J  Feet  (28  in.)  =  A  Military  Pace. 


Tables  of  Weights  and  Measures.  169 

CLOTH  MEASURE. 

TABLE. 

2J  Inches  (in.)  =         1  Nail,  na. 

4     Nails  (9  inches)          =         1  Quarter.  qr. 

4     Quarters  =         1  Yard,  yd. 

This  table  formerly  contained  : 

The  Flemish  Ell,  which  equaled  3  quarters  or  2*7  inches  ; 
The  English  Ell,  which  equaled  5  quarters  or  45  inches  ; 
The  French  Ell,  which  equaled  6  quarters  or  54  inches. 

These  units  of  measure  are  nearly  out  of  use. 


SURVEYORS'  AND  ENGINEERS'  MEASURE. 

TABLE. 

7.92  Inches  =  1  Link,  li. 

25        Links  =  1  Rod  or  Pole,  rd.  or  po. 

4        Poles  or )  .  n,    .  , 

66        Feet         }  =  l  Cham'  ch" 

80        Chains  =  1  Mile,  m. 

Engineers  use  another  chain  which  consists  of  100  links,  each 
1  foot  long. 


SQUARE  OR  SURFACE  MEASURE. 

TABLE. 

144     Square  Inches  (sq.  in.)    =     1  Square  Foot,  sq.  ft. 

9     Square  Feet  =     1  Square  Yard,  sq.  yd. 

301  Square  Yards  }  '  ^J^  £  ^ 

40     Square  Rods  or  Perches  =     1  Rood,  r. 

4     Roods  =     1  Acre,  a. 

640     Acres    1  =     l  Square  Mile,  sq.  m. 

/  or  Section,  sec. 

36     Square  Miles  (6  miles  sq.)=     1  Township,  T. 

16     Perches  =     1  Square  Chain,  sq.  ch. 

10     Square  Chains  =     1   Acre, 


170         Arithmetical  Exercises  and  Examples. 


CUBIC  OR  SOLID  MEASURE. 

TABLE- 


1728     Cubic  Inches 
27     Cubic  Feet 
16     Cubic  Feet 

8     Cord  Feet  or  128  Cubic  Feet 
24J  Cubic  Feet,  or  16J  feet  long, 

1}  high  and  I  foot  wide 
40     Cubic  Feet  of  round  timber, 
50     Cubic  Feet  of  hewn  timber 


I     = 


1  Cubic  Foot. 
1  Cubic  Yard. 
1  Cord  Foot 
1  Cord  of  Wood. 

1    Perch. 

1   Ton  or  Load. 


A  cubic  foot  contains  7  4805  Wine  gallons. 
A  cubic  foot   contains  .2374  barrels     " 
A  cubic  foot  contains  .8082  bushels. 


LIQUID  MEASURE. 

TABLE. 


4  Gills  (gi.) 

2  Pints 

4  Quarts 

31J  Gallons 

2  Barrels  or  63  gallons 

2  Hogsheads 

2  Pipes 


2   Pints  (pt.) 
8  Quarts 
4  Pecks 
8  Bushels 
36  Bushels 


1  Pint,  pt. 

1  Quart,  qt. 

1   Gallon,  gal.  = 

1  Barrel,  bbl. 
1  Hogshead,   hhd. 

1  Pipe,  P. 

1   Tun,  T. 


:231  cubic  in. 


DRY  MEASURE. 


TABLE. 


=  1  Quart, 

=  1  Peck, 

=  1  Bushel, 

=  1  Quarter, 

—  1  Chaldron, 


qt. 
pk. 
bu. 
qr. 
ch. 


TROY  OR  MINT  WEIGHT. 

TABLE. 


24  Grains  (gr.) 
20  Peiiuywcigtits 
12  Ounces 


=     1  Pennyweight, 
^=     1  Ounce, 
*=     1  Pound, 


dwt.  or  pwt. 
oz. 

m. 


Tables  of  Weights  and  Measures.  171 

AVOIRDUPOIS  WEIGHT. 

TABLE. 

Grains  =  1  Dram.                    dr. 

16  Drams  =  1   Ounce,                   oz. 

16  Ounces  —  1   Pound,                   ft). 

25  Pounds  =  1   Quarter,                qr. 

4  Quarters  or  100  pounds  =  1  Hundredweight,  cwt. 

20  Hundredweight  or  2000  pounds=  1   Ton,  t. 

480  Pounds  =  1  Imperial  Quarter. 

100  Pounds  is  also  called  1   Cental,  c. 

The  cwt.  in  England,  and  in  some  cases  in  the  United  States, 
is  112  pounds,  or  4  quarters  of  28  pounds.  The  ton  English  is 
2240  pounds.  Thi,s  is  called  the  long  ton,  and  2000  pounds,  the 
short  ton. 


APOTHECARIES'  WEIGHT. 

TABLE. 


20  Grains  (gr.)  =  1  Scruple, 

3  Scruples  =  1  Dram, 

8  Drams  =  1  Ounce, 

12  Ounces  =  1  Pound, 


MEDICAL  DIVISIONS  OF  THE  GALLON. 

TABLE. 

60  Minims  (m.)  =  1  Fluidram,  fz 

8  Fluidrama  =  1  Fluidounce,  f!$ 

16  Fluidounces  =  1  Pint,  0. 

8  Pints  =  1  Gallon,  Cong. 

0.  is  an  abbreviation   of  octans,  the   Latin  for  one-eighth  ; 
Gong,  for  congiarium,  the  Latin  for  gallon. 


DIAMOND  WEIGHT. 

TABLE. 

16  Parts  =     1     Grain. 

4  Grains  =     1     Carat. 

1  Carat  —     3£  Grain*  Troy,  nearly. 


172        Arithmetical  Exercises  and  Examples. 
ASSAYERS'  WEIGHT. 

TABLE. 

1  Carat  =     10  Pwts.  Troy. 

1   Carat  grain  =       2  Pwts.  12  grains,  or  60  grains  Troy. 
24  Carats  ==       1  Pound  Troy. 

The  term  carat  is  also  used  to  express  the  fineness  of  gold,- 
each  carat  meaning  a  twenty-fourth  part. 


SHOEMAKERS'  MEASURE. 

No.  1  small  size  is  4J  inches,  and  every  succeeding  No.  increases 
^^f  an  inch  to  13. 

No.  1  large  size  is  8JJ  inches,  and  every  succeeding  No.  in- 
creases J  of  an  inch  to  15. 


CIRCULAR  MEASURE. 

TABLE. 

60  Seconds  "  =     1  Minute, 

60  Minutes  =     1  Degree, 

30  Degrees  ==     1  Sign,  s. 

12  Signs  or  360°  =     1   Circle,  c. 

90  degrees  make  1  quadrant  or  right  angle. 
60         "  "      1  sextant  or  sixth  of  a  circle. 

180         u  "      1  semi-circle  or  half-circle. 


MISCELLANEOUS  TABLES. 

BOOKS  AND  PAPER. 

SIZE  OF  PAPER. 
Inches. 


Demy  17  by  22 

Medium  19   "   24 

Double  medium  24   "   38 

Super-royal  21    "   27 


Inches. 


Letter  10  by  15 

Folio  post  16   "    21 

Foolscap  14   "    17 

Crown  15    '•    20 

Double  Elephant  26    "    40 


Imperial  22    "   32 

A  sheet  (medium)  folded  in  2  leaves  is  called  folio. 

"  «  "        4         "         u        quarto  or  4to. 

«  «  "        8         "         "        octavo  or  8vo. 

"  u  u      12         "         "        duodecimo  or  12  mo. 


Tables  of  Weights  and  Measures. 


—     1  Ream. 


1    Bale 


A  sheet  (medium)  folded  in  16  leaves  is  called  16mo. 
«  "  «          18         "  "         18mo. 

»  a  u          24         "  "         24rao. 

«  »<  "          32         •«  "         32mo. 

—       1  Quire. 
=     20  Quires 

=       1   Bundle  ;  5  Bundles    = 
:       1  dozen. 

i     12  dozen     =     1  gross. 
r       1  great  gross. 
1  score. 

-  1   firkin  of  butter. 

=       1  quintal  of  dried  fish. 

-  1   barrel  of  flour. 

=       1  barrel  of  flour  in  California. 
1   barrel  of  beef,  pork,  or  fish. 
=       1   barrel  of  salt. 

1   cask  'of  raisins. 
14  11).  iron  or  lead  =  1   stone. 
}'2  barrels  of  wheat  =  7  English  quarters. 
:u.l   stone=  1   Pig;    8  pigs  =  1  fother. 
256  pounds  of  soap  =  1  barrel. 
25  pounds  of  powder  =  1   keg. 
18  Inches  =  1  Cubit. 


24  Sheets 

480  Sheets 

2  Reams 

12  units 
144  units 
12  gross 
20  units 
56  It). 
100  It). 
106    11.. 
200  tt). 
200   Ib. 
280   II). 
loo  Ib. 


WEIGHT  OK  GRAIN   AND  PRODUCE  PER  BUSHEL, 

AS  USBD  IN    NEW  ORLEANS    WIIBN    THERE  IS  NO    AGREEMENT    TO   T11K 
CONTRARY. 


Wheat,                     bu 

sh.  60  Ib. 

Flaxseed,                 bu 

sh.  56  Ib. 

Corn,                              < 

56    " 

Hempseed, 

44    " 

Rye, 

">6    " 

Buckwheat, 

52     " 

Oats, 

32    " 

Castor  Beans, 

46  -  li 

Barley,                        * 

48    " 

Dried  Peaches, 

33    " 

Irish  Potatoes,           4 

60    " 

Dried  Apples, 

24    " 

Sweet  Potatoes,        ' 

(10     " 

Onions, 

57     " 

Beans,                           ' 

62    " 

Coarse  Salt, 

50    '• 

Bran,                              ' 

24    u 

Fine  Salt, 

50     " 

Clover  seed,               ' 

60    <; 

Stone  Coal, 

80    " 

Timothy  seed.             ' 

45    " 

Corn  Meal, 

44    li 

Barley  Malt.               ' 

34    " 

Plastering  Hair. 

7    4< 

Peas,  split.                  ' 

60      M 

Blue  Grass  seed, 

10    " 

Small  Hominy,          ' 

50    u 

VB - 1 7433 


M306O21 

QA  / 

503 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


Soule's 


. 


College  and  Literary  Institute, 


In  wl 


/*/. 


jj,  /•>/•  Institute, 

For  b« 

f  >///><•'    TwJthif/  SV//oo/, 

There   are 

arious 

fields  of  educqfjpp ;  its  curriculum 

THIRTY-TWO    BRANCHES    OF    STUr  5T, 

taught    by    the    most    improved,    »    ogresr- 
meth 

Lectures  are  given  on  i.he  various  subjects  studied,  an.,  also  on 
-IOLOGY,    HYGIENE,    PURE:  OL- 
ETC. 


our  stn 


GEO.    SOUT-E, 


I*  *0pt 


